Chain Rule Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Find the derivative of f(x)=(x2+1)5f(x) = (x^2 + 1)^5.

Solution

  1. 1
    Outer function: u5u^5, inner function: u=x2+1u = x^2 + 1.
  2. 2
    Chain rule: ddx[u5]=5u4⋅dudx\frac{d}{dx}[u^5] = 5u^4 \cdot \frac{du}{dx}.
  3. 3
    Inner derivative: dudx=2x\frac{du}{dx} = 2x.
  4. 4
    Result: f′(x)=5(x2+1)4⋅2x=10x(x2+1)4f'(x) = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4.

Answer

f′(x)=10x(x2+1)4f'(x) = 10x(x^2 + 1)^4
This is the classic chain rule example. The inner function x2+1x^2 + 1 has derivative 2x2x, which multiplies the result. Without the chain rule, you would need to expand (x2+1)5(x^2+1)^5 first — far more work.

About Chain Rule

The derivative of a composite function f(g(x))f(g(x)) equals f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.

Learn more about Chain Rule →

More Chain Rule Examples

Example 1 easy

Find the derivative of [formula].

Example 3 hard

Find the derivative of [formula].

Example 4 easy

Find the derivative of [formula].

Example 5 medium

Find the derivative of [formula].

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