What is the (cos^3(t))^' ?

The (cos^3(t))^' is -3cos^2(t)sin(t)

What is the first (cos^3(t))^' ?

The first (cos^3(t))^' is -3cos^2(t)sin(t)

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(cos^{3} (t))^{^{'}}

- 3 cos^{2} (t) sin (t)

Solution steps

(cos^{3} (t))^{'}

Apply the chain rule:

3 (cos (t))^{2} (cos (t))^{'}

= 3 (cos (t))^{2} (cos (t))^{'}

(cos (t))^{'} = - sin (t)

= 3 (cos (t))^{2} (- sin (t))

Simplify

= - 3 cos^{2} (t) sin (t)

\frac{d}{d x} (sin (1000 x))

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\frac{d}{d x} ((2 x - 2)^{4} (x^{2} + x + 1)^{5})