How do you find the derivative of $arcsin e^x$ ?

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Answer 1 · 2016-06-23T01:22:16

$e^x/sqrt(1-e^(2x))$

There are two methods:

Using the pre-memorized arcsine derivative:

You may already know that the derivative of arcsine is:

$d/dxarcsin(x)=1/sqrt(1-x^2)$

We can apply the chain rule to this for $arcsin(e^x)$ :

$d/dxarcsin(e^x)=1/sqrt(1-(e^x)^2)*d/dxe^x$

$=e^x/sqrt(1-e^(2x))$

Without knowing the arcsine derivative:

Let

$y=arcsin(e^x)$

Thus:

$sin(y)=e^x$

Differentiate both sides (the chain rule will be used on the left-hand side!):

$y^'*cos(y)=e^x$

$y^'=e^x/cos(y)$

Note that we should express $cos(y)$ in terms of $sin(y)$ , since we know that $sin(y)=e^x$ .

We know that

$sin^2(y)+cos^2(y)=1" "=>" "cos(y)=sqrt(1-sin^2(y))$

$y^'=e^x/sqrt(1-sin^2(y))$

And since $sin(y)=e^x$ :

$y^'=e^x/sqrt(1-(e^x)^2)=e^x/sqrt(1-e^(2x))$

How do you find the derivative of arcsin e^xarcsin e^x?