Chain Rule Math Example 5

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

Example 5

medium
Find the derivative of f(x)=4x+3f(x) = \sqrt{4x + 3}.

Solution

  1. 1
    Rewrite: f(x)=(4x+3)1/2f(x) = (4x + 3)^{1/2}.
  2. 2
    Outer: u1/2u^{1/2}, inner: u=4x+3u = 4x + 3.
  3. 3
    Chain rule: 12(4x+3)−1/2⋅4=24x+3\frac{1}{2}(4x+3)^{-1/2} \cdot 4 = \frac{2}{\sqrt{4x+3}}.

Answer

f′(x)=24x+3f'(x) = \frac{2}{\sqrt{4x + 3}}
Rewriting roots as fractional powers lets you apply the chain rule directly.

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 2 medium

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Example 3 hard

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Example 4 easy

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