Product and quotient rules

Objective. Differentiate products and quotients of functions.

1. Learn
2. Worked example
3. Practice
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Learn: the key idea

For uv the derivative is u′v + uv′. For u/v it is (u′v − uv′)/v². The order matters in the quotient rule.

Worked example

Differentiate y = x²e^x.

1 $u = x², u' = 2x.$
2 $v = e^x, v' = e^x.$
3 $u'v + uv' = 2xe^x + x²e^x = xe^x(2 + x).$

Answer: xe^x(x + 2)

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Common mistake. Reversing the subtraction in the quotient rule numerator.

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