How do you differentiate f(x)=xlnx  using the product rule?

Feb 8, 2016

$f ' \left(x\right) = \ln x + 1$

Explanation:

For $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$, the product rule states that

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

In your case, let $g \left(x\right) = x$ and $h \left(x\right) = \ln x$.

Let's compute the derivatives of $g \left(x\right)$ and $h \left(x\right)$:

$g \left(x\right) = x \text{ " => " } g ' \left(x\right) = 1$

$h \left(x\right) = \ln x \text{ " => " } h ' \left(x\right) = \frac{1}{x}$

Thus, you can compute the derivative as follows:

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

$= 1 \cdot \ln x + x \cdot \frac{1}{x}$

$= \ln x + 1$