Find Derivatives of Functions in Calculus

Comprehensive Differentiation Examples: Power, Product, Quotient, and Chain Rules

Mastering differentiation involves applying specific rules based on the function's structure. Below are examples covering the Power Rule, Product Rule, Quotient Rule, and the Chain Rule with detailed solutions.

Power Rule Examples

The Power Rule states that for \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).

Example 7: Find the derivative of \( f(x) = 4x^5 - 3x^2 + 7x - 10 \)

Apply the power rule to each term individually:

  • The derivative of \( 4x^5 \) is \( 4(5x^4) = 20x^4 \)
  • The derivative of \( -3x^2 \) is \( -3(2x) = -6x \)
  • The derivative of \( 7x \) is \( 7(1) = 7 \)
  • The derivative of the constant \( -10 \) is \( 0 \)

Final Solution: \( f'(x) = 20x^4 - 6x + 7 \)

Example 8: Find the derivative of \( f(x) = \dfrac{3}{x^2} + 5\sqrt[3]{x} \)

Rewrite the function using exponents: \( f(x) = 3x^{-2} + 5x^{1/3} \)

Apply the power rule:

\[ f'(x) = 3(-2x^{-3}) + 5(\frac{1}{3}x^{-2/3}) \] \[ f'(x) = -\dfrac{6}{x^3} + \dfrac{5}{3\sqrt[3]{x^2}} \]

Product and Quotient Rule Examples

Example 1: \( f(x) = (x^2 - 5)(x^3 - 2x + 3) \) (Product Rule)

Using \( f'(x) = U'V + UV' \):

\[ f'(x) = 2x(x^3 - 2x + 3) + (x^2 - 5)(3x^2 - 2) \]

Result: \( 5x^4 - 21x^2 + 6x + 10 \)

Example 3: \( f(x) = \dfrac{x^2 + 1}{5x - 3} \) (Quotient Rule)

Using \( f'(x) = \dfrac{U'V - UV'}{V^2} \):

\[ f'(x) = \dfrac{2x(5x - 3) - (x^2 + 1)5}{(5x - 3)^2} \]

Result: \( \dfrac{5x^2 - 6x - 5}{(5x - 3)^2} \)

Chain Rule Examples

The Chain Rule is used for composite functions: \( [f(g(x))]' = f'(g(x)) \cdot g'(x) \).

Example 9: Find the derivative of \( f(x) = (4x^3 - 2x)^6 \)

Let the "inner" function be \( u = 4x^3 - 2x \). Then \( f(x) = u^6 \).

1. Differentiate the outer function: \( 6u^5 \)

2. Differentiate the inner function: \( u' = 12x^2 - 2 \)

3. Multiply them: \( 6(4x^3 - 2x)^5 \cdot (12x^2 - 2) \)

Result: \( 12(6x^2 - 1)(4x^3 - 2x)^5 \)

Example 10: Find the derivative of \( f(x) = \sqrt{x^2 + 1} \)

Rewrite as \( f(x) = (x^2 + 1)^{1/2} \).

\[ f'(x) = \dfrac{1}{2}(x^2 + 1)^{-1/2} \cdot (2x) \]

The 2s cancel out, leaving:

\[ f'(x) = \dfrac{x}{\sqrt{x^2 + 1}} \]

Practice Exercises

Exercise 7: \( f(x) = 10x^4 - \pi x + e \) (Power Rule)

Answer: \( f'(x) = 40x^3 - \pi \) (Note: \( e \) is a constant, its derivative is 0).

Exercise 8: \( g(x) = (2x+1)^3(x-5)^2 \) (Product + Chain Rule)

Answer: \( g'(x) = 3(2x+1)^2(2)(x-5)^2 + (2x+1)^3(2)(x-5) \)

Simplified: \( 2(2x+1)^2(x-5)[3(x-5) + (2x+1)] = 2(2x+1)^2(x-5)(5x-14) \)

Exercise 9: \( h(x) = \sin(x^2) \) (Trig + Chain Rule)

Answer: \( h'(x) = \cos(x^2) \cdot 2x = 2x\cos(x^2) \)

Further Reading