Hey all on mhf,
Im stuck on a this question and needed some help please.
Differentiate:
$\displaystyle x\sqrt{1-x^2}-sin^{-1}\sqrt{1-x^2}$
Oh dear the LaTex is still not enabled.
I hope you can still see what the question is, Thank you.
Hello, dadon!
That arcsin expression is very messy to differentiate, so I modified it.Differentiate: .f(x) .= .x·(1 - x²)^½ - arcsin(1 - x²)^½
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