1. ## Differentiate

Hey all on mhf,

Im stuck on a this question and needed some help please.

Differentiate:

$x\sqrt{1-x^2}-sin^{-1}\sqrt{1-x^2}$

Oh dear the LaTex is still not enabled.

I hope you can still see what the question is, Thank you.

Differentiate: .f(x) .= .x·(1 - x²)^½ - arcsin(1 - x²)^½
That arcsin expression is very messy to differentiate, so I modified it.

We have: .θ .= .arcsin(1 - x²)^½ . . sin θ .= .(1 - x²)^½

θ is an angle is a right triangle with: .opp = (1 - x²)^½ .and .hyp = 1.
. . Pythagorus tell us that: .adj = x
Hence: .cos θ = x . . θ = arccos(x)

The function becomes: . f(x) .= .x·(1 - x²)^½ - arccos(x)

Then: . f'(x) .= .x·½(1 - x²)^{-½)(-2x) + (1 - x²)^½ + (1 - x²)^{-½)

Rearrange: .(1 - x²)^{-½} - x²·(1 - x²)^{-½} + (1 - x²)^½

Factor: . (1 - x²)·(1 - x²)^{-½} + (1 - x²)^½

. . . . = .(1 - x²)^½ + (1 - x²)^½

. . . . = .2(1 - x²)^½