What is the derivative of #2^arcsin(x)#?

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Answer 1 · 2016-08-03T16:26:05

#(2^arcsin(x)ln(2))/sqrt(1-x^2)#

Let:

#y=2^arcsin(x)#

Take the natural logarithm of both sides:

#ln(y)=ln(2^arcsin(x))#

Simplify using logarithm rules:

#ln(y)=arcsin(x)*ln(2)#

Differentiate both sides. You should remember that:

The left-hand side will need the chain rule, similar to implicit differentiation.
On the right-hand side, #ln(2)# is just a constant.
The derivative of #arcsin(x)# is #1/sqrt(1-x^2)#.

Differentiating yields:

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

Now, solving for #dy/dx#, the derivative, multiply both sides by #y#.

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

Write #y# as #2^arcsin(x)#:

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