What is the derivative of #arcsin(x) + xsqrt(1-x^2)#?

1 Answer
Jun 8, 2015

By the knowledge of the derivative of arcsine, the Product Rule, and the Chain Rule, we can write

#d/dx(arcsin(x)+x\sqrt{1-x^{2}})=\frac{1}{sqrt{1-x^2}}+\sqrt{1-x^{2}}+\frac{1}{2}x(1-x^{2))^{-1/2}*(-2x)#.

Simplification gives

#d/dx(arcsin(x)+x\sqrt{1-x^{2}})=\frac{1+(1-x^2)-x^{2}}{sqrt{1-x^2}}#

#=\frac{2(1-x^2)}{sqrt{1-x^2}}=2\sqrt{1-x^2}#

The interesting thing about this example is that it implies

#\int sqrt{1-x^2}\ dx=1/2arcsin(x)+1/2x\sqrt{1-x^{2}}+C#.