Derivative of Inverse Trigonometric Functions

Question

f(x) = sin(arccos(4x))

What is f'(x)?

Answer 1

f(x)= sin(arccos(4x)) = sqrt(1 - 16x^2)
(To prove that, draw yourself a triangle with cos A = 4x and figure out the sin of A)

let u = 1 - 16x^2
f(u) = sqrt u
f'(x) = df/dx = df/du du/dx
= (1/2)(u)^-1/2 * -32 x
= -16 x/sqrt[1 - 16x^2]

Answer 2

Thanks. The next question I have is pretty similar.

f(x) =cos(arcsin(2x))

Find f'(x)

Answer 3

It is so similar that you should be able to do it the same way. Draw the triangle and you have sqrt(1-4x^2)
f(x) = (1-4x^2)^.5
f'(x) = .5 [(1-4x^2)^-.5] (-8x)
= -4x/sqrt(1-4x^2)