Question

Find the derivative of y = sin^3(2x * tan^2(2x)) with respect to x, given that dy/dx = π/8.

A) (3/8)sin^2(2x * tan^2(2x))cos(2x * tan^2(2x))
B) (π/8)sin^2(2x * tan^2(2x))cos(2x * tan^2(2x))
C) (3/8)sin^2(2x * tan^2(2x))cos^2(2x * tan^2(2x))
D) (π/8)sin^2(2x * tan^2(2x))cos^2(2x * tan^2(2x))

Answer 1

The derivative of y = sin^3(2x \cdot tan^2(2x)) should be found using the chain and product rules. The provided answer choices do not match the outcome of this differentiation process.

Step-by-step explanation:

The problem asks us to find the derivative of the function y = sin^3(2x \cdot tan^2(2x)) with respect to x. We need to apply the chain rule multiple times to differentiate this function, taking into account the composition of functions involved in y.

First, let us consider the outermost function, which is sin^3(u), where u = 2x \cdot tan^2(2x). Its derivative, by the chain rule, is 3sin^2(u) \cdot cos(u) times the derivative of u. The derivative of u with respect to x combines the product rule and the chain rule.

The given information that dy/dx = π/8 does not apply directly to this problem, as this appears to be additional information not required for solving this particular derivative.

None of the provided answer choices correctly matches the derivative of the given function y. The correct answer would need to include the detailed derivative calculation using the product and chain rules. The answer choices given appear to relate to a different problem or contain errors.