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

Question

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

affwysaumlgx · Accepted Answer

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), the derivative is as follows:

F' (x) = f' (g (x)) (g' (x))

Or, in words:
= the derivative of the outer function multiplied by the derivative of the inner function.
Thus, let's take a look at the question.

y = \tan (3 x)

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 = 3 x

, so the 3x is composed in the tan. Deriving, we get:
The derivative of the outer function (without considering the inside function):

\frac{d}{d x} \tan (3 x) = \sec^{2} (3 x)

The derivative of the inner function:

\frac{d}{d x} 3 x = 3

Combining, we get:

\frac{d}{d x} y = y' = 3 \sec^{2} (3 x)

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

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