differentiate, please show work

label Calculus
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Apr 29th, 2015

r = (sin^-1 (- 6x^4))^2

To take this derivative, you need to use the chain rule - so, you take the derivative of the function outside of parentheses first (the raising of the part in parentheses to 2) using the power rule, and then you perform the derivative for the inside of the function (the arcsine of -6x^4). You also need to know that the derivative of sin^-1 (x) = 1/(sqrt(1-x^2).

Using the power rule, the derivative of the outer function is:  2(sin^-1 (-6x^4))

The derivative of (sin^-1 (-6x^4)) is 1/(sqrt(1-(-6x^4)^2) * (4)(-6)x^3 - this is the derivative of the arcsine compenent, multiplied by the derivative of (-6x^4)^2.

Putting it all together, we have:

dr/dx = [2(sin^-1 (-6x^4) )]*[1/(sqrt(1-(-6x^4)^2) * (4)(-6)x^3]

Simplifying, we get:

dr/dx = 2(sin^-1 (-6x^4) )*1/(sqrt(1 - 36x^8)) * (-24)x^3

dr/dx = (-48)x^3 (sin^-1 (-6x^4))/ (sqrt(1 - 36x^8)

The answer is D.

Apr 29th, 2015

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