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

1 Answer
Aug 3, 2016

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

Explanation:

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)#