Product Rule

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


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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