How do you take the derivative of # tan(2x)#?

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Answer 1 · 2015-09-05T18:40:17

Using the chain rule #tan(2x)=2sec^2(2x)#

The chain rule is as follows:

#d/dx f(g(x)) = [d/dx f(x)]|_(x=g(x)) * d/dx g(x)#

...or, in words,
1) Get the derivative of the outer function, plug in the inner function...
2) ...multiplied by the derivative of the inner function.

In #tan(2x)#, the outer function is #tan x# and the inner function is #2x#.

The derivative of #tan x# is #sec^2 x#.
Plug in #2x#, and we have #sec^2 (2x)#.

So, after our first step, we have:
#d/dx tan(2x)#
#=sec^2 (2x) * d/dx (2x)#

Then, we continue:
#d/dx tan(2x)#
#=sec^2 (2x) * d/dx (2x)#

#=sec^2 (2x) * (2)#

#=2sec^2(2x)#