Question

Use the chain rule of differentiation to find the derivative with respect to t of g(t)=cos(ωt) .

Answer 1

To find the derivative of g(t)=cos(ωt) with respect to t using the chain rule, we differentiate the outer function (cosine) and multiply by the derivative of the inner function (ωt), which results in g'(t) = -ω sin(ωt).

Step-by-step explanation:

The student has asked how to use the chain rule to find the derivative with respect to t of the function g(t) = cos(ωt). The chain rule is a formula for computing the derivative of the composition of two or more functions. In this case, we have an outer function which is the cosine and an inner function which is the product of ω (a constant) and t (the variable).

To apply the chain rule, we differentiate the outer function (cosine) with respect to its argument and then multiply by the derivative of the inner function with respect to t. The derivative of cos(u), where u is a function of t, is -sin(u), and the derivative of ωt with respect to t is ω.

Therefore, the derivative of g(t) with respect to t is:

g'(t) = -ω sin(ωt).

Answer 2

To differentiate g(t) = cos(ωt), apply the chain rule: differentiate the cosine function to get -sin(u), then differentiate the inner function ωt to get ω; multiply both derivatives to obtain d/dt of g(t) = -ωsin(ωt).

Step-by-step explanation:

To find the derivative of g(t) = cos(ωt) with respect to t using the chain rule, follow these steps:

• Identify the outer function and the inner function. In this case, g(t) has the outer function cosine and the inner function ωt.
• First, differentiate the outer function with respect to its argument. The derivative of cos(u) with respect to u is -sin(u).
• Then, differentiate the inner function (ωt) with respect to t. The derivative of ωt with respect to t is just ω.
• Multiply the derivatives of the outer and inner functions to obtain the total derivative.

So, d/dt of g(t) = -ωsin(ωt).

The negative sign comes from the derivative of the cosine function, and ω is the constant factor from the derivative of the inner function.