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



Answered question


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

Answer & Explanation



Beginner2023-02-22Added 12 answers

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):
d d x tan ( 3 x ) = sec 2 ( 3 x )
The derivative of the inner function:
d d x 3 x = 3
Combining, we get:
d d x y = y = 3 sec 2 ( 3 x )

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