What is the derivative of arccot(e^2x)?

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February 25th 2009, 07:07 AM
Kaitosan

What is the derivative of arccot(e^2x)?

What is the derivative of arccot(e^2x)? I did this problem by switching the expression to arctan[e^(-2x)] then got (-2e^-2x)/(e^-4x + 1) but my book says (-2e^2x)/(1 + e^4x).

What am I doing wrong?
February 25th 2009, 08:00 AM
Thomas154321

Let $y=\cot^{-1}(e^{2x})$ , then $\cot(y)=e^{2x}, \tan(y)=e^{-2x}$

Now differentiate both sides wrt x:

$\sec^2(y)\frac{dy}{dx}=-2e^{-2x}$ and so $\frac{dy}{dx}=\frac{-2e^{-2x}}{\sec^2(y)}$

Using $\sec^2(y)=1+\tan^2(y) = 1+e^{-4x}$ we have

$\frac{dy}{dx}=\frac{-2e^{-2x}}{1+e^{-4x}}$

Now multiply top and bottom by e^{4x} to get $\frac{-2e^{2x}}{e^{4x}+1}$ .
February 25th 2009, 08:37 AM
Abu-Khalil

Also, $y=\text{arccot}\left(e^{2x}\right)=\arctan\left(e^ {-2x}\right)$ $\Rightarrow y'=\frac{-2e^{-2x}}{e^{-4x}+1}=\frac{-2e^{2x}}{e^{4x}+1}$ .
February 25th 2009, 08:54 AM
Kaitosan

Hey Abu-Khalil! Thanks a lot, I don't know why I didn't recognize the fact that I could've simplified the expression by multiplying the numerator and the denominator by e^4x. At least I technically answered correctly. (Talking)

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