# How do you differentiate f(x)=ln2x * tan3x using the product rule?

Mar 25, 2016

I found: $f ' \left(x\right) = \tan \frac{3 x}{x} + \frac{3 \ln \left(2 x\right)}{\cos} ^ 2 \left(3 x\right)$

#### Explanation:

The Product Rule tells us that if we a function $f$ that is the product of 2 other functions:
$f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$
differentiating we get:
$f ' \left(x\right) = g ' \left(x\right) h \left(x\right) + g \left(x\right) h ' \left(x\right)$
in our case:
$f ' \left(x\right) = \frac{2}{2 x} \tan \left(3 x\right) + \ln \left(2 x\right) \frac{1}{\cos} ^ 2 \left(3 x\right) \cdot 3$
Where I used the Chain Rule to differentiate each component function;
finally:
$f ' \left(x\right) = \tan \frac{3 x}{x} + \frac{3 \ln \left(2 x\right)}{\cos} ^ 2 \left(3 x\right)$