How do you find the derivative of y=tan(3x) ?

Aug 31, 2014

Well, you could do this using the chain rule, since there is a function within a function (a "composite" function). The chain rule is:

If you have a composite function F(x), then the derivative is:

$F ' \left(x\right) = f ' \left(g \left(x\right)\right) \left(g ' \left(x\right)\right)$

Or, in words:

=the derivative of the outer function with the inside function left alone times the derivative of the inner function.

So let's look at your question.
$y = \tan \left(3 x\right)$

The outer function is tan and the inner function is $3 x$, since $3 x$ is "inside" the tan. Think of it as $\tan \left(u\right)$ where $u = 3 x$, so the $3 x$ is composed in the tan. Deriving, we get:

The derivative of the outer function (leaving the inside function alone):
$\frac{d}{\mathrm{dx}} \tan \left(3 x\right) = {\sec}^{2} \left(3 x\right)$
The derivative of the inner function:
$\frac{d}{\mathrm{dx}} 3 x = 3$
Combining, we get:
$\frac{d}{\mathrm{dx}} y = y ' = 3 {\sec}^{2} \left(3 x\right)$