poprskanvxcl

2023-02-21

How to find the derivative of $$?

affwysaumlgx

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:

$$

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.

$$

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

The derivative of the outer function (without considering the inside function):

$$

The derivative of the inner function:

$$

Combining, we get:

$$

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

$$

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.

$$

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

The derivative of the outer function (without considering the inside function):

$$

The derivative of the inner function:

$$

Combining, we get:

$$