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!

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