# Product Rule

The derivative of a function f \left( x \right) = u \left( x \right) v \left( x \right) is $$f' \left( x \right) = u' \left( x \right) v \left( x \right) + u \left( x \right) v' \left( x \right)$$

## Example

Let f \left( x \right) = x^2 \ln \left( x \right).

It is made of 2 functions: u \left( x \right) = x^2 and v \left( x \right) = \ln \left( x \right). We know their derivatives:

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

The function u \left( x \right) = \ln \left( x \right) is the logarithmic function. It's derivative is: \begin{align*} u' \left( x \right) &= \frac{1}{x} \end{align*} So the derivative of f \left( x \right) is \begin{align*} f' \left( x \right) &= u' \left( x \right) v \left( x \right) + u \left( x \right) v' \left( x \right) \\ &= 2x \ln \left( x \right) + x^2 \frac{1}{x} \\ &= 2x \ln \left( x \right) + x \\ \end{align*} 

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