# What is the derivative of f(x)=x*ln(x) ?

Jul 23, 2014

The function $f \left(x\right) = x \cdot \ln \left(x\right)$ is of the form $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$ which makes it suitable for appliance of the product rule.

Product rule says that to find the derivative of a function that's a product of two or more functions use the following formula:

$f ' \left(x\right) = g ' \left(x\right) h \left(x\right) + g \left(x\right) h ' \left(x\right)$

In our case, we can use the following values for each function:

$g \left(x\right) = x$

$h \left(x\right) = \ln \left(x\right)$

$g ' \left(x\right) = 1$

$h ' \left(x\right) = \frac{1}{x}$

When we substitute each of these into the product rule, we get the final answer:

$f ' \left(x\right) = 1 \cdot \ln \left(x\right) + x \cdot \frac{1}{x} = \ln \left(x\right) + 1$