Chain Rule Math Example 3

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Example 3

hard
Find the derivative of f(x)=sin⁡(x3)f(x) = \sin(x^3).

Solution

  1. 1
    Outer function: sin⁡(u)\sin(u), inner function: u=x3u = x^3.
  2. 2
    Chain rule: ddx[sin⁡(u)]=cos⁡(u)⋅dudx\frac{d}{dx}[\sin(u)] = \cos(u) \cdot \frac{du}{dx}.
  3. 3
    Inner derivative: dudx=3x2\frac{du}{dx} = 3x^2.
  4. 4
    Result: f′(x)=cos⁡(x3)⋅3x2=3x2cos⁡(x3)f'(x) = \cos(x^3) \cdot 3x^2 = 3x^2\cos(x^3).

Answer

f′(x)=3x2cos⁡(x3)f'(x) = 3x^2 \cos(x^3)
The chain rule applies identically to trigonometric compositions. The derivative of sin⁡\sin is cos⁡\cos, but you must multiply by the derivative of the argument x3x^3.

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

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Example 2 medium

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Example 4 easy

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Example 5 medium

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