# Quotient Rule

The derivative of a function f \left( x \right) = \frac{u \left( x \right)}{v \left( x \right)} is $$f' \left( x \right) = \frac{u' \left( x \right) v \left( x \right) - u \left( x \right) v' \left( x \right)}{v \left( x \right)^{2}}$$

## Example

Let f \left( x \right) = \frac{x^3}{\ln \left( x \right)}. It is the quotient of 2 functions: u \left( x \right) = x^3 and v \left( x \right) = \ln \left( x \right).

The function u \left( x \right) = x^{3} is a power function x^{n} with n=3. Use the power rule \begin{align*} u' \left( x \right) &= 3 x ^{3-1} \\ &= 3 x^{2} \end{align*}

The function v \left( x \right) = \ln \left( x \right) is the logarithmic function. It's derivative is know: \begin{align*} v' \left( x \right) = \frac{1}{x} \end{align*}

The derivative of f \left( x \right) is \begin{align*} f' \left( x \right) &= \frac{u' \left( x \right) v \left( x \right) - u \left( x \right) v' \left( x \right)}{v \left( x \right)^{2}} \\ &= \frac{3x^2 \ln \left( x \right) - x^{3} \frac{1}{x}}{\ln \left( x \right)^{2}} \\ &= \frac{3x^2 \ln \left( x \right) - x^{2}}{\ln \left( x \right)^{2}} \\ \end{align*}

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