What is the derivative of #y=4xsin^-1(x)+4sqrt(1-x^2)#?

1 Answer
Alvin L.
Jan 2, 2018

#d/dx(4xsin^-1(x)4sqrt(1-x^2))=4sin^-1(x)#

Explanation:

First, I'll split the derivative up into two:
#d/dx(4xsin^-1(x))+d/dx(4sqrt(1-x^2))#

I'll call the left one derivative 1 and the right one derivative 2.

Derivative 1
Here we'll use the product rule:
#d/dx(4xsin^-1(x))=d/dx(4x)*sin^-1(x)+4x*d/dx(sin^-1(x))#

#d/dx(4x)=4#

#d/dx(sin^-1(x))=1/sqrt(1-x^2)#

#d/dx(4xsin^-1(x))=4sin^-1(x)+(4x)/sqrt(1-x^2)#

Derivative 2
#d/dx(4sqrt(1-x^2))=4*d/dx(sqrt(1-x^2))=#

This time we can use the chain rule. I'll let #u=1-x^2#:
#=4*d/(du)(sqrtu)*d/dx(1-x^2)#

#d/dx(sqrtx)=1/(2sqrtx)#

#d/dx(1-x^2)=-2x#

#d/dx(4sqrt(1-x^2))=4*1/(2sqrtu)*-2x=#

We can resubstitute with #u=1-x^2#:
#=2/sqrt(1-x^2)*-2x=#

#=-(4x)/sqrt(1-x^2)#

Completing the original derivative
Now that we know Derivative 1 and Derivative 2, we can complete the original derivative:
#d/dx(4xsin^-1(x)4sqrt(1-x^2))=#

#=4sin^-1(x)+(4x)/sqrt(1-x^2)-(4x)/sqrt(1-x^2)=#

#=4sin^-1(x)#

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