How To Use The Quotient Rule For Derivatives - Math Warehouse

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Example 1

Suppose $$\displaystyle f(x) = \frac{2x+3}{5x + 1}$$. Find $$f'(x)$$.

Step 1

Differentiate using the Quotient Rule. Parts in $$\blue{blue}$$ are related to the numerator $$\blue{2x+3}$$

$$ f'(x) = \frac{(5x+1)\cdot \blue 2 - \blue{(2x+3)}\cdot 5}{(5x+1)^2} $$

Step 2

Simplify the numerator.

$$ \begin{align*} f'(x) & = \frac{2(5x+1) - 5(2x+3)}{(5x+1)^2}\\[6pt] & = \frac{10x+2 - 10x-15}{(5x+1)^2}\\[6pt] & = \frac{-13}{(5x+1)^2} \end{align*} $$

Answer

$$\displaystyle f'(x) = -\frac{13}{(5x+1)^2}$$

Note: There is no reason to expand the denominator. In fact, doing so will make the derivative harder to work with.

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