Question

Find f'(x) for f(x)=cos^2(5x^3).

Answer 1

d/dx cos^2(5x^3)

= d/dx [cos(5x^3)]^2

= 2[cos(5x^3)]

= - 2[cos(5x^3)] * sin(5x^3)

= - 2[cos(5x^3)] * sin(5x^3) * 15x^2

= - 30[cos(5x^3)] * sin(5x^3) * x^2

Explanation:

d/dx x^n = nx^(n - 1)

d/dx cos x = - sin x

Chain rule:

d/dx f(g(...w(x))) = f’(g(...w(x))) * g’(...w(x)) * ... * w’(x)

Answer 2

Step-by-step explanation:

f(x) = cos²(5x³)

f(x) = (cos(5x³))²

Use chain rule to find the derivative:

f'(x) = 2 cos(5x³) × -sin(5x³) × 15x²

f'(x) = -30x² sin(5x³) cos(5x³)

If desired, use double angle formula to simplify:

f'(x) = -15x² sin(10x³)