Differentiation Rules Math Example 3

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Example 3

easy
Differentiate f(x)=x2cos⁡xf(x) = x^2 \cos x using the product rule.

Solution

  1. 1
    Let u=x2u = x^2, v=cos⁡xv = \cos x, so u′=2xu' = 2x, v′=−sin⁡xv' = -\sin x.
  2. 2
    Product rule: f′(x)=2xcos⁡x+x2(−sin⁡x)=2xcos⁡x−x2sin⁡xf'(x) = 2x\cos x + x^2(-\sin x) = 2x\cos x - x^2\sin x.

Answer

f′(x)=2xcos⁡x−x2sin⁡xf'(x) = 2x\cos x - x^2\sin x
Apply the product rule term by term. The derivative of cos⁡x\cos x is −sin⁡x-\sin x, so the second term picks up a negative sign.

About Differentiation Rules

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

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More Differentiation Rules Examples

Example 1 easy

Use the product rule to differentiate [formula].

Example 2 medium

Use the quotient rule to differentiate [formula].

Example 4 hard

Differentiate [formula] using the quotient rule.

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