Final answer:
To find the derivative of y = tan(sin(3x)), we apply the chain rule. This involves finding the derivatives of both the outer function tan(u) and the inner function u=sin(3x), resulting in the derivative of y with respect to x: 3sec^2(sin(3x))cos(3x).
Step-by-step explanation:
The question is asking to find the derivative of a function y = tan(sin(3x)) with respect to x. To find this derivative, we would use the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
The step-by-step process involves taking the derivative of tan(u), where u = sin(3x), which gives us sec2(u). We then differentiate the inner function u, yielding 3cos(3x). Finally, we multiply the derivatives together for the result: d/dx(y) = 3sec2(sin(3x))cos(3x).