Question

Use the chain rule of differentiation to find the derivative with respect to t of g(t) = cos(wt). dg/dt = ∘ -wtsin(wt) ∘ -wsin(wt) ∘ 0 ∘ -sin(wt) ∘ wcos(wt)

Answer 1

The derivative of g(t) = cos(wt) with respect to t using the chain rule is -w sin(wt), which involves differentiating the outer cosine function and multiplying it by the derivative of the inner function wt.

Step-by-step explanation:

To find the derivative with respect to t of g(t) = cos(wt), we use the chain rule of differentiation. The chain rule states that if a function y is composed of a function u which itself is a function of v, i.e., y = f(u(v)), then the derivative of y with respect to v is dy/dv = (df/du) * (du/dv). In our case, the outer function is the cosine function and the inner function is wt.

Applying the chain rule:

1. The derivative of the outer function with respect to the inner function cos(u) is -sin(u).
2. The derivative of the inner function wt with respect to t is w.

Therefore, dg(t)/dt = d(cos(wt))/dt = -sin(wt) * w.

Comparing with the given options, the correct answer is -w sin(wt).