# How do you use the product rule to find the derivative of y=x*ln(x) ?

Mar 29, 2018

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

#### Explanation:

$y = x \cdot \ln \left(x\right)$

The product rule states that for a function $f \left(x\right) = g \left(x\right) h \left(x\right)$, the derivative is:

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

In our equation, we have:

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

We compute the derivatives:

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

We use the product rule:

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

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

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