Using the quotient rule to find the derivative

The quotient rule differentiates a quotient, just like the product rule differentiates a product

Just as you must always use the product rule when two variable expressions are multiplied, you must use the quotient rule whenever two variable expressions are divided.

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Given a function

$h(x)=\frac{f(x)}{g(x)}$

then its derivative is

$h'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{\left[g(x)\right]^2}$

Applying the quotient rule formula to find the derivative

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Using quotient rule with power and log functions

Example

Use quotient rule to find the derivative.

$h(x)=\frac{x^2}{\ln{x}}$

Based on the quotient rule formula, we know that $f(x)$ is the numerator and therefore $f(x)=x^2$ and that $g(x)$ is the denominator and therefore that $g(x)=\ln{x}$. $f'(x)=2x$, and $g'(x)=1/x$.

Just as you must always use the product rule when two variable expressions are multiplied, you must use the quotient rule whenever two variable expressions are divided.

Plugging all of these components into the quotient rule gives

$h'(x)=\frac{\left(\ln{x}\right)(2x)-\left(x^2\right)\left(\frac{1}{x}\right)}{\left(\ln{x}\right)^2}$

$h'(x)=\frac{2x\ln{x}-x}{\left(\ln{x}\right)^2}$

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