Differentiation Rules Math Example 1

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

Example 1

easy
Use the product rule to differentiate f(x)=x3⋅sin⁡xf(x) = x^3 \cdot \sin x.

Solution

  1. 1
    Identify the two factors: u=x3u = x^3 and v=sin⁡xv = \sin x.
  2. 2
    Find their derivatives: u′=3x2u' = 3x^2 and v′=cos⁡xv' = \cos x.
  3. 3
    Apply the product rule (uv)′=u′v+uv′(uv)' = u'v + uv'.
  4. 4
    Result: f′(x)=3x2sin⁡x+x3cos⁡xf'(x) = 3x^2 \sin x + x^3 \cos x.

Answer

f′(x)=3x2sin⁡x+x3cos⁡xf'(x) = 3x^2 \sin x + x^3 \cos x
The product rule states (fg)′=f′g+fg′(fg)' = f'g + fg'. Each factor is differentiated once while the other is kept intact, and the two results are added. Never multiply the individual derivatives together.

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 2 medium

Use the quotient rule to differentiate [formula].

Example 3 easy

Differentiate [formula] using the product rule.

Example 4 hard

Differentiate [formula] using the quotient rule.

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