50 Practice Problems: Power Rule, Sum Rule & Elementary Functions Differentiation Exercises

Introduction

Differentiation forms the backbone of calculus, and mastering the fundamental rules is essential for success in mathematics, engineering, and physics courses. These 50 practice exercises focus specifically on three core differentiation techniques that every student must master: the power rule, sum rule, and derivatives of elementary functions.

Whether you’re preparing for your next calculus exam, working through homework assignments, or simply want to strengthen your mathematical foundation, these problems provide the targeted practice you need. Each exercise builds upon the concepts covered in our comprehensive Lecture 4: Mastering Basic Differentiation Rules: Power Rule, Sum Rule, and Elementary Functions, giving you practical application of the theoretical concepts.

The problems range from straightforward applications of single rules to more complex scenarios that require combining multiple differentiation techniques. You’ll encounter polynomial functions, radical expressions, exponential and logarithmic functions, and trigonometric expressions – all designed to reinforce your understanding and build confidence in your problem-solving abilities.

Students often struggle with differentiation because they lack sufficient practice with the basic rules. These exercises address that gap by providing structured problems that progress logically from simple to more challenging scenarios. Each problem type appears multiple times with different variations, ensuring you develop both speed and accuracy in your calculations.

Take your time working through each problem, and don’t hesitate to review the corresponding lecture material when you encounter difficulties. Consistent practice with these fundamental rules will prepare you for more advanced calculus topics like the product rule, quotient rule, and chain rule.

50 Comprehensive Practice Exercises: Power Rule, Sum Rule & Elementary Functions Differentiation Exercises

Basic Level (Problems 1-15)

Focus: Direct application of the power rule, sum rule, and basic elementary function derivatives

Power Rule Applications

1. Find f'(x) if f(x) = x^7

2. Find the derivative of g(x) = x^4

3. Differentiate h(x) = x^(-3)

4. Find d/dx[x^(1/3)]

5. If f(x) = √x, find f'(x)

Constants and Linear Functions

6. Find the derivative of f(x) = 12

7. Differentiate g(x) = 5x

8. Find f'(x) if f(x) = -3x + 7

Basic Sum Rule

9. Find the derivative of f(x) = x^3 + x^2

10. Differentiate g(x) = 2x^4 – 3x^2 + 5

11. Find f'(x) if f(x) = x^5 – 4x + 1

Elementary Functions

12. Find the derivative of f(x) = e^x

13. Differentiate g(x) = ln(x)

14. Find f'(x) if f(x) = sin(x)

15. Find the derivative of h(x) = cos(x)

Intermediate Level (Problems 16-35)

Focus: Combining multiple rules, mixed functions, and basic applications

Mixed Polynomial Functions

16. Find f'(x) if f(x) = 3x^4 – 2x^3 + x^2 – 5x + 8

17. Differentiate g(x) = x^6 – 4x^(-2) + 7x^(1/2)

18. Find the derivative of h(x) = 2x^3 – 1/x^2 + √x

19. If f(x) = (x^2 + 3x – 1)/x, rewrite and find f'(x)

20. Find f'(x) if f(x) = x^(3/2) – 2x^(-1/2) + 4

Elementary Functions with Coefficients

21. Differentiate f(x) = 3e^x – 2ln(x) + 4

22. Find g'(x) if g(x) = 5sin(x) + 2cos(x)

23. Find the derivative of h(x) = 4tan(x) – sec(x)

24. Differentiate f(x) = 2e^x + 3ln(x) – x^2

25. Find f'(x) if f(x) = -3sin(x) + 4cos(x) + x^3

Negative and Fractional Exponents

26. Find the derivative of f(x) = x^(-5) + x^(2/3)

27. Differentiate g(x) = 3/x^4 – 2√[3]{x}

28. Find f'(x) if f(x) = 1/x^3 + x^(3/4) – 5

29. Find the derivative of h(x) = 2x^(-1/2) + 4x^(5/2)

30. Differentiate f(x) = x^(-2/3) – 3x^(4/3) + 1

Basic Applications

31. The position of a particle is given by s(t) = t^3 – 4t^2 + 2t. Find the velocity function v(t).

32. If the cost function is C(x) = 0.01x^2 + 5x + 100, find the marginal cost function.

33. A population grows according to P(t) = 1000e^t. Find the rate of population growth.

34. The height of a projectile is h(t) = -16t^2 + 64t + 80. Find the velocity function.

35. If revenue is R(x) = 50x – 0.1x^2, find the marginal revenue function.

Advanced Level (Problems 36-45)

Focus: Complex combinations, inverse functions, advanced applications, and problem-solving

Complex Mixed Functions

36. Find f'(x) if f(x) = x^4 – 2e^x + 3ln(x) – 5sin(x) + cos(x)

37. Differentiate g(x) = 2x^(3/2) – 4/x^3 + e^x – tan(x)

38. Find the derivative of h(x) = 3x^(-2/3) + 2ln(x) – sin(x) + 4x^(5/4)

39. If f(x) = (2x^3 – 1)/x^2 + e^x – cos(x), find f'(x)

40. Differentiate f(x) = x^π – π^x + πe^x (treat π as a constant)

Inverse Trigonometric Functions

41. Find the derivative of f(x) = arcsin(x) + arctan(x)

42. Differentiate g(x) = 2arccos(x) – 3arcsin(x)

43. Find f'(x) if f(x) = x^2 + arctan(x) – ln(x)

Advanced Applications

44. The temperature of a cooling object follows T(t) = 25 + 75e^(-0.1t). Find the rate of temperature change.

45. A company’s profit is P(x) = -0.001x^3 + 0.5x^2 – 10x – 5000. Find the marginal profit function and determine when the rate of profit change is zero.

Challenge Problems (Problems 46-50)

Focus: Integration of concepts, optimization setup, and real-world complex scenarios

46. Multi-variable rate problem: If the radius of a circle is changing at a rate given by r(t) = 2t + 1, and the area formula is A = πr^2, find dA/dr and explain what this represents physically.

47. Optimization setup: A rectangle has one side along the x-axis and two vertices on the curve y = 16 – x^2. If the rectangle has width 2x, find the derivative of the area function A(x) = 2x(16 – x^2).

48. Economics application: A monopolist faces a demand curve p(x) = 100 – 0.5x and has a cost function C(x) = x^2 + 10x + 50. The profit function is π(x) = xp(x) – C(x) = x(100 – 0.5x) – (x^2 + 10x + 50). Find π'(x) and determine the critical points.

49. Related rates setup: A ladder 25 feet long leans against a wall. If the bottom slides away at 3 ft/s, the height h and base distance b are related by h = √(625 – b^2). Find dh/db and interpret its meaning.

50. Composite application: A particle moves along a curve where its x-coordinate is given by x(t) = t^3 – 6t^2 + 9t and its y-coordinate by y(t) = 2t^2 – 8t + 6. Find dx/dt and dy/dt, then determine when the particle is momentarily at rest (both derivatives equal zero).

Answer Key Guidelines

Difficulty Distribution:

• Basic (1-15): Direct rule application – 30%
• Intermediate (16-35): Combined rules and basic applications – 40%
• Advanced (36-45): Complex functions and advanced applications – 20%
• Challenge (46-50): Integration of concepts and real-world scenarios – 10%

Skills Assessed:

• ✓ Power Rule mastery
• ✓ Sum and Difference Rules
• ✓ Elementary function derivatives
• ✓ Function combination and simplification
• ✓ Real-world application setup
• ✓ Mathematical communication
• ✓ Problem-solving strategies

50 Comprehensive Practice Exercises: Answer Key

Answer Key with Detailed Solutions

Basic Level Solutions (Problems 1-15)

Focus: Direct application of the power rule, sum rule, and basic elementary function derivatives

Problem 1: Basic Power Rule Application

Find f'(x) if f(x) = x^7

Technique Used: Power Rule for polynomial functions

Step-by-Step Solution:

1. Identify the function form: f(x) = x^n where n = 7
2. Apply the power rule: d/dx[x^n] = nx^(n-1)
3. Substitute n = 7: f'(x) = 7x^(7-1)
4. Simplify the exponent: f'(x) = 7x^6

Answer: f'(x) = 7x^6

Problem 2: Power Rule with Different Exponent

Find the derivative of g(x) = x^4

Technique Used: Power Rule for polynomial functions

Step-by-Step Solution:

1. Identify the function form: g(x) = x^n where n = 4
2. Apply the power rule: d/dx[x^n] = nx^(n-1)
3. Substitute n = 4: g'(x) = 4x^(4-1)
4. Simplify: g'(x) = 4x^3

Answer: g'(x) = 4x^3

Problem 3: Negative Exponent Power Rule

Differentiate h(x) = x^(-3)

Technique Used: Power Rule with negative exponents

Step-by-Step Solution:

1. Identify the function form: h(x) = x^n where n = -3
2. Apply the power rule: d/dx[x^n] = nx^(n-1)
3. Substitute n = -3: h'(x) = (-3)x^(-3-1)
4. Simplify the exponent: h'(x) = -3x^(-4)
5. Alternative form: h'(x) = -3/x^4

Answer: h'(x) = -3x^(-4) or h'(x) = -3/x^4

Problem 4: Fractional Exponent Power Rule

Find d/dx[x^(1/3)]

Technique Used: Power Rule with fractional exponents

Step-by-Step Solution:

1. Identify the function form: f(x) = x^n where n = 1/3
2. Apply the power rule: d/dx[x^n] = nx^(n-1)
3. Substitute n = 1/3: f'(x) = (1/3)x^(1/3-1)
4. Simplify the exponent: 1/3 – 1 = 1/3 – 3/3 = -2/3
5. Final form: f'(x) = (1/3)x^(-2/3)
6. Alternative form: f'(x) = 1/(3x^(2/3)) = 1/(3∛(x²))

Answer: f'(x) = (1/3)x^(-2/3) or f'(x) = 1/(3∛(x²))

Problem 5: Square Root Function

If f(x) = √x, find f'(x)

Technique Used: Power Rule with radical expressions converted to fractional exponents

Step-by-Step Solution:

1. Rewrite the radical: √x = x^(1/2)
2. Apply the power rule: d/dx[x^n] = nx^(n-1) where n = 1/2
3. Calculate: f'(x) = (1/2)x^(1/2-1)
4. Simplify the exponent: 1/2 – 1 = -1/2
5. Result: f'(x) = (1/2)x^(-1/2)
6. Convert back to radical form: f'(x) = 1/(2√x)

Answer: f'(x) = (1/2)x^(-1/2) or f'(x) = 1/(2√x)

Problem 6: Constant Function

Find the derivative of f(x) = 12

Technique Used: Derivative of a constant

Step-by-Step Solution:

1. Recognize that f(x) = 12 is a constant function
2. Apply the constant rule: d/dx[c] = 0 for any constant c
3. Since 12 is a constant, its derivative is 0

Answer: f'(x) = 0

Problem 7: Linear Function with Coefficient

Differentiate g(x) = 5x

Technique Used: Constant multiple rule and power rule

Step-by-Step Solution:

1. Recognize the form: g(x) = cx where c = 5
2. Apply constant multiple rule: d/dx[cf(x)] = c·f'(x)
3. Since f(x) = x, we have f'(x) = 1
4. Therefore: g'(x) = 5 · 1 = 5

Answer: g'(x) = 5

Problem 8: Linear Function with Constant Term

Find f'(x) if f(x) = -3x + 7

Techniques Used: Sum rule, constant multiple rule, and constant rule

Step-by-Step Solution:

1. Apply sum rule: d/dx[f(x) + g(x)] = f'(x) + g'(x)
2. Break down: f(x) = -3x + 7
3. Differentiate each term:
   • d/dx[-3x] = -3 · d/dx[x] = -3 · 1 = -3
   • d/dx[7] = 0 (constant rule)
4. Combine results: f'(x) = -3 + 0 = -3

Answer: f'(x) = -3

Problem 9: Simple Polynomial

Find the derivative of f(x) = x^3 + x^2

Technique Used: Sum rule and power rule

Step-by-Step Solution:

Problem 10: Polynomial with Coefficients

Differentiate g(x) = 2x^4 – 3x^2 + 5

Techniques Used: Sum rule, constant multiple rule, power rule, and constant rule

Step-by-Step Solution:

Problem 11: Mixed Polynomial

Find f'(x) if f(x) = x^5 – 4x + 1

Techniques Used: Sum rule, constant multiple rule, power rule, and constant rule

Step-by-Step Solution:

Problem 12: Exponential Function

Find the derivative of f(x) = e^x

Technique Used: Standard derivative of natural exponential function

Step-by-Step Solution:

Problem 13: Natural Logarithm

Differentiate g(x) = ln(x)

Technique Used: Standard derivative of natural logarithm

Step-by-Step Solution:

Problem 14: Sine Function

Find f'(x) if f(x) = sin(x)

Technique Used: Standard derivative of the sine function

Step-by-Step Solution:

Problem 15: Cosine Function

Find the derivative of h(x) = cos(x)

Technique Used: Standard derivative of the cosine function

Step-by-Step Solution:

Intermediate Level Solutions (Problems 16-35)

Focus: Combining multiple rules, mixed functions, and basic applications

Problem 16: Complex Polynomial

Find f'(x) if f(x) = 3x^4 – 2x^3 + x^2 – 5x + 8

Techniques Used: Sum rule, constant multiple rule, power rule, and constant rule

Step-by-Step Solution:

1. Apply the sum/difference rule to each term
2. Differentiate each term:
   • d/dx[3x^4] = 3 · 4x^3 = 12x^3
   • d/dx[-2x^3] = -2 · 3x^2 = -6x^2
   • d/dx[x^2] = 2x
   • d/dx[-5x] = -5
   • d/dx[8] = 0
3. Combine all terms: f'(x) = 12x^3 – 6x^2 + 2x – 5

Answer: f'(x) = 12x^3 – 6x^2 + 2x – 5

Problem 17: Mixed Exponents

Differentiate g(x) = x^6 – 4x^(-2) + 7x^(1/2)

Technique Used: Power rule with positive, negative, and fractional exponents

Step-by-Step Solution:

1. Apply the power rule to each term:
   • d/dx[x^6] = 6x^5
   • d/dx[-4x^(-2)] = -4 · (-2)x^(-3) = 8x^(-3)
   • d/dx[7x^(1/2)] = 7 · (1/2)x^(-1/2) = (7/2)x^(-1/2)
2. Combine results: g'(x) = 6x^5 + 8x^(-3) + (7/2)x^(-1/2)
3. Alternative form: g'(x) = 6x^5 + 8/x^3 + 7/(2√x)

Answer: g'(x) = 6x^5 + 8x^(-3) + (7/2)x^(-1/2)

Problem 18: Radical and Negative Exponent Mix

Find the derivative of h(x) = 2x^3 – 1/x^2 + √x

Technique Used: Power rule with conversion of radicals and fractions to exponential form

Step-by-Step Solution:

1. Rewrite in exponential form: h(x) = 2x^3 – x^(-2) + x^(1/2)
2. Apply the power rule to each term:
   • d/dx[2x^3] = 2 · 3x^2 = 6x^2
   • d/dx[-x^(-2)] = -(-2)x^(-3) = 2x^(-3)
   • d/dx[x^(1/2)] = (1/2)x^(-1/2)
3. Combine: h'(x) = 6x^2 + 2x^(-3) + (1/2)x^(-1/2)
4. Alternative form: h'(x) = 6x^2 + 2/x^3 + 1/(2√x)

Answer: h'(x) = 6x^2 + 2x^(-3) + (1/2)x^(-1/2)

Problem 19: Rational Function Simplification

If f(x) = (x^2 + 3x – 1)/x, rewrite and find f'(x)

Technique Used: Algebraic simplification followed by the power rule

Step-by-Step Solution:

1. Rewrite by dividing each term: f(x) = x^2/x + 3x/x – 1/x
2. Simplify: f(x) = x + 3 – x^(-1)
3. Apply the power rule to each term:
   • d/dx[x] = 1
   • d/dx[3] = 0
   • d/dx[-x^(-1)] = -(-1)x^(-2) = x^(-2)
4. Combine: f'(x) = 1 + 0 + x^(-2) = 1 + x^(-2)
5. Alternative form: f'(x) = 1 + 1/x^2

Answer: f'(x) = 1 + x^(-2) or f'(x) = 1 + 1/x^2

Problem 20: Fractional Exponents

Find f'(x) if f(x) = x^(3/2) – 2x^(-1/2) + 4

Technique Used: Power rule with fractional and negative fractional exponents

Step-by-Step Solution:

1. Apply the power rule to each term:
   • d/dx[x^(3/2)] = (3/2)x^(3/2-1) = (3/2)x^(1/2)
   • d/dx[-2x^(-1/2)] = -2 · (-1/2)x^(-1/2-1) = x^(-3/2)
   • d/dx[4] = 0
2. Combine: f'(x) = (3/2)x^(1/2) + x^(-3/2)
3. Alternative form: f'(x) = (3/2)√x + 1/(x^(3/2)) = (3/2)√x + 1/(x√x)

Answer: f'(x) = (3/2)x^(1/2) + x^(-3/2)

Problem 21: Mixed Elementary Functions

Differentiate f(x) = 3e^x – 2ln(x) + 4

Technique Used: Standard derivatives of exponential and logarithmic functions with the constant multiple rule

Step-by-Step Solution:

1. Apply derivatives to each term:
   • d/dx[3e^x] = 3 · e^x = 3e^x
   • d/dx[-2ln(x)] = -2 · (1/x) = -2/x
   • d/dx[4] = 0
2. Combine results: f'(x) = 3e^x – 2/x + 0

Answer: f'(x) = 3e^x – 2/x

Problem 22: Trigonometric Functions

Find g'(x) if g(x) = 5sin(x) + 2cos(x)

Technique Used: Standard derivatives of trigonometric functions with the constant multiple rule

Step-by-Step Solution:

1. Apply derivatives to each term:
   • d/dx[5sin(x)] = 5 · cos(x) = 5cos(x)
   • d/dx[2cos(x)] = 2 · (-sin(x)) = -2sin(x)
2. Combine results: g'(x) = 5cos(x) – 2sin(x)

Answer: g'(x) = 5cos(x) – 2sin(x)

Problem 23: Advanced Trigonometric Functions

Find the derivative of h(x) = 4tan(x) – sec(x)

Technique Used: Standard derivatives of tangent and secant functions

Step-by-Step Solution:

1. Apply derivatives to each term:
   • d/dx[4tan(x)] = 4 · sec^2(x) = 4sec^2(x)
   • d/dx[-sec(x)] = -sec(x)tan(x)
2. Combine results: h'(x) = 4sec^2(x) – sec(x)tan(x)

Answer: h'(x) = 4sec^2(x) – sec(x)tan(x)

Problem 24: Mixed Functions

Differentiate f(x) = 2e^x + 3ln(x) – x^2

Technique Used: Combination of exponential, logarithmic, and polynomial derivatives

Step-by-Step Solution:

1. Apply derivatives to each term:
   • d/dx[2e^x] = 2e^x
   • d/dx[3ln(x)] = 3 · (1/x) = 3/x
   • d/dx[-x^2] = -2x
2. Combine results: f'(x) = 2e^x + 3/x – 2x

Answer: f'(x) = 2e^x + 3/x – 2x

Problem 25: Complex Mixed Function

Find f'(x) if f(x) = -3sin(x) + 4cos(x) + x^3

Technique Used: Trigonometric and polynomial derivatives combined

Step-by-Step Solution:

Problem 26: Negative and Fractional Exponents

Find the derivative of f(x) = x^(-5) + x^(2/3)

Technique Used: Power rule with negative and fractional exponents

Step-by-Step Solution:

Problem 27: Cube Root and Reciprocal

Differentiate g(x) = 3/x^4 – 2∛x

Technique Used: Power rule with conversion to exponential form

Step-by-Step Solution:

Problem 28: Mixed Negative Exponents

Find f'(x) if f(x) = 1/x^3 + x^(3/4) – 5

Technique Used: Power rule with exponential form conversion

Step-by-Step Solution:

Problem 29: Square Root Variations

Find the derivative of h(x) = 2x^(-1/2) + 4x^(5/2)

Technique Used: Power rule with fractional exponents

Step-by-Step Solution:

Problem 30: Complex Fractional Exponents

Differentiate f(x) = x^(-2/3) – 3x^(4/3) + 1

Technique Used: Power rule with negative and positive fractional exponents

Step-by-Step Solution:

Problem 31: Velocity from Position

The position of a particle is given by s(t) = t^3 – 4t^2 + 2t. Find the velocity function v(t).

Technique Used: Physical application of derivative as rate of change (velocity is derivative of position)

Step-by-Step Solution:

Problem 32: Marginal Cost

If the cost function is C(x) = 0.01x^2 + 5x + 100, find the marginal cost function.

Technique Used: Economic application of derivative (marginal cost is derivative of total cost)

Step-by-Step Solution:

Problem 33: Population Growth Rate

A population grows according to P(t) = 1000e^t. Find the rate of population growth.

Technique Used: Exponential growth model with derivative representing rate of change

Step-by-Step Solution:

Problem 34: Projectile Velocity

The height of a projectile is h(t) = -16t^2 + 64t + 80. Find the velocity function.

Technique Used: Physics application with quadratic motion model

Step-by-Step Solution:

Problem 35: Marginal Revenue

If revenue is R(x) = 50x – 0.1x^2, find the marginal revenue function.

Technique Used: Economic application of marginal analysis

Step-by-Step Solution:

Advanced Level Solutions (Problems 36-45)

Focus: Complex combinations, inverse functions, advanced applications, and problem-solving

Problem 36: Complex Mixed Function

Find f'(x) if f(x) = x^4 – 2e^x + 3ln(x) – 5sin(x) + cos(x)

Technique Used: Integration of all basic differentiation rules

Step-by-Step Solution:

1. Apply the appropriate derivative to each term:
   • d/dx[x^4] = 4x^3
   • d/dx[-2e^x] = -2e^x
   • d/dx[3ln(x)] = 3/x
   • d/dx[-5sin(x)] = -5cos(x)
   • d/dx[cos(x)] = -sin(x)
2. Combine all terms: f'(x) = 4x^3 – 2e^x + 3/x – 5cos(x) – sin(x)

Answer: f'(x) = 4x^3 – 2e^x + 3/x – 5cos(x) – sin(x)

Problem 37: Advanced Mixed Function

Differentiate g(x) = 2x^(3/2) – 4/x^3 + e^x – tan(x)

Technique Used: Combination of fractional exponents, negative exponents, exponential, and trigonometric functions

Step-by-Step Solution:

1. Rewrite: g(x) = 2x^(3/2) – 4x^(-3) + e^x – tan(x)
2. Apply derivatives:
   • d/dx[2x^(3/2)] = 2(3/2)x^(1/2) = 3x^(1/2) = 3√x
   • d/dx[-4x^(-3)] = -4(-3)x^(-4) = 12x^(-4) = 12/x^4
   • d/dx[e^x] = e^x
   • d/dx[-tan(x)] = -sec^2(x)
3. Combine: g'(x) = 3√x + 12/x^4 + e^x – sec^2(x)

Answer: g'(x) = 3x^(1/2) + 12x^(-4) + e^x – sec^2(x)

Problem 38: Very Complex Mixed Function

Find the derivative of h(x) = 3x^(-2/3) + 2ln(x) – sin(x) + 4x^(5/4)

Technique Used: Advanced combination of fractional exponents, logarithmic, and trigonometric functions

Step-by-Step Solution:

1. Apply derivatives to each term:
   • d/dx[3x^(-2/3)] = 3(-2/3)x^(-5/3) = -2x^(-5/3)
   • d/dx[2ln(x)] = 2/x
   • d/dx[-sin(x)] = -cos(x)
   • d/dx[4x^(5/4)] = 4(5/4)x^(1/4) = 5x^(1/4)
2. Combine: h'(x) = -2x^(-5/3) + 2/x – cos(x) + 5x^(1/4)

Answer: h'(x) = -2x^(-5/3) + 2/x – cos(x) + 5x^(1/4)

Problem 39: Rational Function with Mixed Terms

If f(x) = (2x^3 – 1)/x^2 + e^x – cos(x), find f'(x)

Technique Used: Algebraic simplification followed by mixed differentiation

Step-by-Step Solution:

1. Simplify the rational part: (2x^3 – 1)/x^2 = 2x – x^(-2)
2. Rewrite: f(x) = 2x – x^(-2) + e^x – cos(x)
3. Apply derivatives:
   • d/dx[2x] = 2
   • d/dx[-x^(-2)] = -(-2)x^(-3) = 2x^(-3)
   • d/dx[e^x] = e^x
   • d/dx[-cos(x)] = sin(x)
4. Combine: f'(x) = 2 + 2x^(-3) + e^x + sin(x)
5. Alternative form: f'(x) = 2 + 2/x^3 + e^x + sin(x)

Answer: f'(x) = 2 + 2x^(-3) + e^x + sin(x)

Problem 40: Function with π as Constant

Differentiate f(x) = x^π – π^x + πe^x (treat π as a constant)

Technique Used: Power rule with constant exponent, exponential with constant base, and constant multiple

Step-by-Step Solution:

1. Apply derivatives treating π as a constant:
   • d/dx[x^π] = πx^(π-1) (power rule)
   • d/dx[-π^x] = -π^x ln(π) (exponential with constant base)
   • d/dx[πe^x] = π · e^x = πe^x (constant multiple of e^x)
2. Combine: f'(x) = πx^(π-1) – π^x ln(π) + πe^x

Answer: f'(x) = πx^(π-1) – π^x ln(π) + πe^x

Problem 41: Inverse Trigonometric Functions

Find the derivative of f(x) = arcsin(x) + arctan(x)

Technique Used: Standard derivatives of inverse trigonometric functions

Step-by-Step Solution:

Problem 42: Multiple Inverse Trigonometric Functions

Differentiate g(x) = 2arccos(x) – 3arcsin(x)

Technique Used: Constant multiples of inverse trigonometric derivatives

Step-by-Step Solution:

Problem 43: Mixed Inverse Trigonometric and Other Functions

Find f'(x) if f(x) = x^2 + arctan(x) – ln(x)

Technique Used: Combination of polynomial, inverse trigonometric, and logarithmic derivatives

Step-by-Step Solution:

Problem 44: Exponential Decay Application

The temperature of a cooling object follows T(t) = 25 + 75e^(-0.1t). Find the rate of temperature change.

Technique Used: Exponential decay model with chain rule concepts

Step-by-Step Solution:

Problem 45: Profit Optimization Setup

A company’s profit is P(x) = -0.001x^3 + 0.5x^2 – 10x – 5000. Find the marginal profit function and determine when the rate of profit change is zero.

Technique Used: Economic application with cubic function and critical point analysis

Step-by-Step Solution:

Challenge Problems Solutions (Problems 46-50)

Focus: Integration of concepts, optimization setup, and real-world complex scenarios

Problem 46: Multi-variable Rate Problem

If the radius of a circle is changing at a rate given by r(t) = 2t + 1, and the area formula is A = πr^2, find dA/dr and explain what this represents physically.

Technique Used: Related rates setup with geometric application

Step-by-Step Solution:

1. Given: A = πr^2, we need dA/dr
2. Differentiate A with respect to r:
   • dA/dr = d/dr[πr^2] = π · 2r = 2πr
3. Physical interpretation: dA/dr represents the rate of change of area with respect to radius
4. This tells us how much the area increases for each unit increase in radius
5. Since dA/dr = 2πr, the rate depends on the current radius – larger circles have faster area growth per unit radius change

Answer: dA/dr = 2πr; This represents the instantaneous rate of area change per unit change in radius

Problem 47: Optimization Setup

A rectangle has one side along the x-axis and two vertices on the curve y = 16 – x^2. If the rectangle has width 2x, find the derivative of the area function A(x) = 2x(16 – x^2).

Technique Used: Product expansion followed by polynomial differentiation

Step-by-Step Solution:

1. Expand the area function: A(x) = 2x(16 – x^2) = 32x – 2x^3
2. Apply the power rule to each term:
   • d/dx[32x] = 32
   • d/dx[-2x^3] = -6x^2
3. Combine: A'(x) = 32 – 6x^2
4. Physical interpretation: A'(x) represents the rate of change of area with respect to half-width x
5. Critical points occur when A'(x) = 0: 32 – 6x^2 = 0, so x^2 = 32/6 = 16/3, thus x = 4/√3

Answer: A'(x) = 32 – 6x^2

Problem 48: Economics Application

A monopolist faces a demand curve p(x) = 100 – 0.5x and has a cost function C(x) = x^2 + 10x + 50. The profit function is π(x) = xp(x) – C(x) = x(100 – 0.5x) – (x^2 + 10x + 50). Find π'(x) and determine the critical points.

Technique Used: Economic optimization with profit maximization

Step-by-Step Solution:

Problem 49: Related Rates Setup

A ladder 25 feet long leans against a wall. If the bottom slides away at 3 ft/s, the height h and base distance b are related by h = √(625 – b^2). Find dh/db and interpret its meaning.

Technique Used: Related rates with Pythagorean relationship and chain rule concepts

Step-by-Step Solution:

Problem 50: Composite Application

A particle moves along a curve where its x-coordinate is given by x(t) = t^3 – 6t^2 + 9t and its y-coordinate by y(t) = 2t^2 – 8t + 6. Find dx/dt and dy/dt, then determine when the particle is momentarily at rest.

Technique Used: Parametric motion analysis with velocity components

Step-by-Step Solution:

Conclusion

Working through these 50 differentiation problems gives you the solid foundation needed for advanced calculus topics. The power rule, sum rule, and elementary function derivatives appear in virtually every calculus course, making these exercises valuable for long-term success.

Students who complete these practice problems typically see improved performance on exams and homework assignments. The repetition helps build the muscle memory needed for quick, accurate calculations during timed tests. More importantly, understanding these basic rules makes complex differentiation techniques much easier to grasp later.

These exercises cover the most common problem types you’ll encounter in calculus courses. From simple polynomial derivatives to combinations involving exponential and trigonometric functions, the variety ensures you’re prepared for different question formats. Many students find that practicing these fundamentals reduces anxiety about calculus exams.

The differentiation skills developed through these problems extend beyond mathematics courses. Engineering students use these techniques in physics, dynamics, and circuit analysis. Business students apply derivatives in optimization problems and economic modeling. Science majors rely on differentiation for rate calculations and data analysis.

Regular practice with basic differentiation rules builds confidence and speed. Students who master these fundamentals early in their calculus journey often find subsequent topics more manageable. The time invested in these practice problems pays dividends throughout your mathematical education.

Keep these exercises handy for review before major exams. The combination of power rule applications, sum rule problems, and elementary function derivatives represents the core skills every calculus student needs. Master these basics, and you’ll be ready for whatever calculus throws at you next.

Key Takeaways from This Practice Set

🎯 Mathematical Mastery Achieved:

• Power Rule proficiency for polynomial and radical functions
• Sum Rule applications for complex multi-term expressions
• Elementary function differentiation (exponential, logarithmic, trigonometric)
• Algebraic manipulation techniques for derivative calculations
• Foundation skills for advanced differentiation rules

🔧 Engineering Applications Mastered:

• Rate of change calculations in mechanical systems
• Velocity and acceleration problems in dynamics
• Electrical circuit analysis using derivative concepts
• Optimization foundations for engineering design
• Signal analysis applications in communications engineering

Next Steps in Your Calculus Journey

Having mastered basic differentiation rules, you’re now prepared for:

1. Product and Quotient Rules – Handle derivatives of function products and ratios
2. Chain Rule Applications – Differentiate composite functions with confidence
3. Implicit Differentiation – Solve equations where y cannot be isolated
4. Related Rates Problems – Apply derivatives to real-world changing quantities
5. Optimization Applications – Find maximum and minimum values in engineering contexts

Share Your Success

Did these practice problems help you master the power rule, sum rule, and elementary function derivatives? Share your experience in the comments below and help fellow engineering students on their calculus journey!

Stay Connected

Follow PinoyBIX.org for more:

✅ Engineering mathematics tutorials and practice problems

✅ Board exam preparation materials

✅ Step-by-step solutions for complex engineering problems

✅ Free resources for Filipino engineering students

Remember: Mathematics is the language of engineering – and you’ve just strengthened your vocabulary in one of its most essential chapters!

Keep practicing, keep learning, and keep building the mathematical foundation that will power your engineering success!

Looking Ahead: From Basic Rules to Advanced Techniques

Now that you’ve mastered the fundamental differentiation rules through these practice problems, you’re ready to tackle more sophisticated derivative calculations that involve products and quotients of functions.

Coming Next: Lecture 5 – The Product and Quotient Rules

Our next lecture will expand your differentiation toolkit with two powerful rules that handle complex function combinations:

• Product Rule Mastery – Learn to differentiate products like (3x² + 1)(sin x) and (e^x)(ln x)
• Quotient Rule Applications – Master derivatives of fractions such as (x³ + 2x)/(cos x) and (tan x)/(x² – 1)
• Strategic Rule Selection – Discover when to use the product rule vs. quotient rule for maximum efficiency
• Engineering Applications – Apply these rules to real-world problems in circuits, mechanics, and optimization
• Common Mistake Prevention – Avoid typical errors that trip up students during exams

These advanced rules build directly on the power rule, sum rule, and elementary function derivatives you’ve just practiced. The combination of these techniques will give you the tools to differentiate virtually any function you encounter in engineering mathematics.

Get ready to take your calculus skills to the next level!