What is the derivative of arcsin(x) + xsqrt(1-x^2)arcsin(x)+x√1−x2?

1 Answer
Bill K.
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)ddx(arcsin(x)+x√1−x2)=1√1−x2+√1−x2+12x(1−x2)−12⋅(−2x).

Simplification gives

d/dx(arcsin(x)+x\sqrt{1-x^{2}})=\frac{1+(1-x^2)-x^{2}}{sqrt{1-x^2}}ddx(arcsin(x)+x√1−x2)=1+(1−x2)−x2√1−x2

=\frac{2(1-x^2)}{sqrt{1-x^2}}=2\sqrt{1-x^2}=2(1−x2)√1−x2=2√1−x2

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.

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