Question

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

Answer 1

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`.