What is the derivative of $f(x)=arctan(2^x)$ ?

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Answer 1 · 2018-04-11T14:01:38

$(2^xln2)/(2^(2x)+1)$

Apply the chain rule:

$dy/dx=dy/(du)*(du)/dx$

Let $u=2^x,:.(du)/dx=2^x*ln2$

Then, $y=arctanu,:.dy/(du)=1/(u^2+1)$

And so,

$dy/dx=1/(u^2+1)*2^x*ln2$

$=(2^xln2)/(u^2+1)$

Undo the substitution to get:

$=(2^xln2)/((2^x)^2+1)$

$=(2^xln2)/(2^(2x)+1)$

What is the derivative of f(x)=arctan(2^x)f(x)=arctan(2^x)?