Question

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

Answer 1

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'(x)=f'(g(x)) (g'(x))`

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 (3x)`

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

The derivative of the outer function (leaving the inside function alone):
`d/dx tan(3x)=sec^2(3x)`
The derivative of the inner function:
`d/dx 3x=3`
Combining, we get:
`d/dx y=y'= 3sec^2(3x)`