calculus q # 3?

Question

calculate the derivative of :

i) sin^(-1) squareroot of (1-x)

ii) tan^(-1) squareroot of (x^2 -1)

iii) (x^2 -1)^(1/x)

Answer 1

Only one problem per question please.

Calculate the derivative of :

i) arcsin[√(1 - x)]

Use the chain rule.
Let u = √(1 - x)

d[arcsin(u)] / du = 1 / √(1 - u²)
And
du/dx = -1 / [2√(1 - x)]

Now put it together.

d{arcsin[√(1 - x)]} / dx = {1 / √(1 - u²)} * (du/dx)

= {1 / √(1 - u²)} * {-1 / [2√(1 - x)]}

= {1 / √[1 - (1 - x)]} * {-1 / [2√(1 - x)]}

= {1 / √x} * {-1 / [2√(1 - x)]}

= -1 / [2√(x - x²)]