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

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)

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)