Calculus: Differentiation

I am having trouble solving these. Could you explain how to do these 3 questions, step by step? That would be helpful!

1. Differentiate y=sin ^-1 (x^2) at x=0.5 . Round to 3 decimal points.

3. Differentiate y= (x^2 +1) tan ^-1 (x) at x=1

7. Differentiate y= [cos^-1 (x)]^3 at x=0.5

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asked by Paige
  1. 1. y = arcsin(x ²)
    Let u = x ²
    y = arcsin(u)
    dy/dx = (dy/du)(du/dx)
    dy/dx = (1/√ (1-u ² ))(2x)
    dy/dx = (1/√ (1-(x ²) ² ))(2x)
    dy/dx = 2x/√ (1-x^4)
    At x = 0.5:
    dy/dx = 2(0.5)/√ (1-0.5^4) ~ 1.033

    Similar approach for 2. and 3, assuming you have the basic inverse trig derivative formulae.

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    posted by Motorhead

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