Finding the derivative of a composite function with chain rule

by Kurious Fox

The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. Here are some examples that illustrate its use:

Example 1: Simple Composite Function

Let’s find the derivative of h(x) = (3x^2 + 2x)^{10} .

  1. Identify the outer function and the inner function:
  • Outer function: f(u) = u^{10}
  • Inner function: g(x) = 3x^2 + 2x
  1. Differentiate the outer function with respect to the inner function:
    f'(u) = 10u^9
  2. Differentiate the inner function with respect to x :
    g'(x) = 6x + 2
  3. Apply the chain rule:
    h'(x) = f'(g(x)) \cdot g'(x) = 10(3x^2 + 2x)^9 \cdot (6x + 2)

Example 2: Trigonometric Function

Find the derivative of y = \sin(5x^3 + 4x) .

  1. Identify the outer function and the inner function:
  • Outer function: f(u) = \sin(u)
  • Inner function: g(x) = 5x^3 + 4x
  1. Differentiate the outer function with respect to the inner function:
    f'(u) = \cos(u)
  2. Differentiate the inner function with respect to x :
    g'(x) = 15x^2 + 4
  3. Apply the chain rule:
    y' = f'(g(x)) \cdot g'(x) = \cos(5x^3 + 4x) \cdot (15x^2 + 4)

Example 3: Exponential Function

Find the derivative of z = e^{\sin(x)} .

  1. Identify the outer function and the inner function:
  • Outer function: f(u) = e^u
  • Inner function: g(x) = \sin(x)
  1. Differentiate the outer function with respect to the inner function:
    f'(u) = e^u
  2. Differentiate the inner function with respect to x :
    g'(x) = \cos(x)
  3. Apply the chain rule:
    z' = f'(g(x)) \cdot g'(x) = e^{\sin(x)} \cdot \cos(x)

Example 4: Logarithmic Function

Find the derivative of w = \ln(2x^2 + 3) .

  1. Identify the outer function and the inner function:
  • Outer function: f(u) = \ln(u)
  • Inner function: g(x) = 2x^2 + 3
  1. Differentiate the outer function with respect to the inner function:
    f'(u) = \frac{1}{u}
  2. Differentiate the inner function with respect to x :
    g'(x) = 4x
  3. Apply the chain rule:
    w' = f'(g(x)) \cdot g'(x) = \frac{1}{2x^2 + 3} \cdot 4x = \frac{4x}{2x^2 + 3}

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