Differentiate

cos
3
⁡
4
𝑥

Question

Differentiate

cos
3
⁡
4
𝑥

BlackTom AI · Accepted Answer

We begin by restating the problem: differentiate cos^3(4x).

Option given: -12 cos^2(4x) sin(4x).

To differentiate cos^3(4x), apply the chain rule in two steps. First, treat cos^3(4x) as [cos(4x)]^3. The derivative of [f(x)]^n is n [f(x)]^(n-1) * f'(x).

Here, f(x) = cos(4x) and n = 3. So the outer differentiation gives 3[cos(4x)]^2, and we must multiply by the derivative of the inner function f(x) = cos(4x).

The derivative of cos(4x) with respect to x is -sin(4x) times the inner derivative of 4x, which is 4. Therefore, f'(x) = -sin(4x) * 4 = -4 sin(4x).

Multiplying these factors together yields: 3[cos(4x)]^2 * (-4 sin(4x)) = -12 cos^2(4x) sin(4x).

Thus, the expression -12 cos^2(4x) sin(4x) matches the result obtained by applying the chain rule to [cos(4x)]^3.

If there were additional options, we would compare each one by one, checking whether they correctly apply the chain rule, include the proper power reduction, and account for the inner derivative of 4x. Any option missing the factor of 4 from the inner derivative or the correct power reduction would be incorrect due to incomplete application of the chain rule or misplacement of the derivative terms.

In summary, the derivative arises from differentiating the outer cube, then differentiating the inner cosine, and multiplying the results: 3[cos(4x)]^2 * (-sin(4x)) * 4 = -12 cos^2(4x) sin(4x).

Differentiate cos 3 ⁡ 4 𝑥

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