How do you find the derivative of #y=cos^-1(e^(2x))#?

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Answer 1 · 2016-10-22T20:49:06

Let's start by determining the derivative of #y = cos^-1(x)#.

#y = cos^-1(x)#

#cosy = x#

Through implicit differentiation, you should have:

#-siny(dy/dx) = 1#

#dy/dx = -1/siny#

Since #siny = sqrt(1 - cos^2y)#, we can rewrite as:

#dy/dx= -1/sqrt(1 - cos^2y)#

Since #x = cosy#, we can substitute:

#dy/dx= -1/sqrt(1 - x^2)#

Now, let's differentiate #e^(2x)#. The derivative of any function of the form #e^f(x)# is #f'(x) xx e^f(x)#. So, the derivative is #2e^(2x)#.

By the chain rule, letting #y = cos^-1(u)# and #u= e^(2x)#.

#dy/dx = -1/sqrt(1 - u^2) xx 2e^(2x)#

#dy/dx = -(2e^(2x))/sqrt(1 - e^(2x))#

Hopefully this helps!