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Skill: Differentiate products using the product rule

Explanation and Free Practice Resources

For a product of two differentiable functions, the derivative is the sum of the two terms formed by differentiating one factor at a time while keeping the other unchanged: (fg)′=f′g+fg′(fg)'=f'g+fg'. This captures how both factors contribute to the rate of change and prevents the misconception that (fg)′=f′g′(fg)'=f'g'; applications include algebraic, trigonometric, exponential, and logarithmic products, while generalized many-factor formulas and advanced theoretical treatments are not included.

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Detailed Explanation: Differentiate products using the product rule

When a function is written as a product of two functions, use the product rule:

(fg)′=f′g+fg′(fg)'=f'g+fg'

This means:

  1. Differentiate the first factor and leave the second unchanged.
  2. Add the first factor unchanged multiplied by the derivative of the second factor.

Do not multiply the two derivatives together.

Example

Differentiate

y=x2sin⁡x.y=x^2\sin x.

Identify the two factors:

f(x)=x2andg(x)=sin⁡x.f(x)=x^2 \qquad\text{and}\qquad g(x)=\sin x.

Differentiate each factor:

f′(x)=2xandg′(x)=cos⁡x.f'(x)=2x \qquad\text{and}\qquad g'(x)=\cos x.

Apply the product rule:

y′=f′g+fg′y'=f'g+fg'

Substitute the functions and their derivatives:

y′=(2x)(sin⁡x)+(x2)(cos⁡x).y'=(2x)(\sin x)+(x^2)(\cos x).

Therefore,

y′=2xsin⁡x+x2cos⁡x\boxed{y'=2x\sin x+x^2\cos x}

Both terms are needed because both factors, x2x^2 and sin⁡x\sin x, are changing.

Learn by doing: Differentiate products using the product rule

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