Calculate the derivative of the following functions.
y = cos^7/4(4x³)

Verified step by step guidance

1. Step 1: Recognize that the function y = \cos^{7/4}(4x^3) is a composition of functions, which requires the use of the chain rule for differentiation.
2. Step 2: Let u = 4x^3. Then, the function becomes y = (\cos(u))^{7/4}. Differentiate y with respect to u using the power rule: \frac{dy}{du} = \frac{7}{4}(\cos(u))^{3/4}(-\sin(u)).
3. Step 3: Differentiate u = 4x^3 with respect to x: \frac{du}{dx} = 12x^2.
4. Step 4: Apply the chain rule: \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. Substitute the expressions from Steps 2 and 3 into this formula.
5. Step 5: Simplify the expression obtained in Step 4 to get the derivative of the original function y with respect to x.