Evaluate the derivative of the following function?

Question

f(w) = cos[sin^-1(5w)]

Answer 1

f(w) = cos[sin^-1(5w)]

Apply 'derivative with respect to w' each side.

f'(w) = (d/dx){cos[sin^-1(5w)]}

Derivative of Trigonometric Functions: (d/dx)[cos(x)] = - sin(x)

f'(w) = - sin[sin^-1(5w)][sin^-1(5w)]'                                              [inner derivative of sin^-1(5w) ]

Derivative of Inverse Trigonometric Functions: (d/dx)sin^-1x = 1/√(1 - x²)

f'(w) = - sin[sin^-1(5w)][1/√(1 - (5w)²)][5w]'                               [inner derivative of (5w) ]

General Derivative Formulas:(d/dx)(x) = 1.

f'(w) = - sin[sin^-1(5w)][1/√(1 - 25w²)][5]

f'(w) = - 5sin[sin^-1(5w)] / √(1 - 25w²).

Answer 2

 

Apply 'derivative with respect to w' on each side.

Apply chain rule in derivatives.

Derivative of Inverse Trigonometric Functions:

Derivative of Inverse Trigonometric Functions: