Final answer:
To find the derivative of the function f(x) = x³ tan(sin(x))², use the chain rule. The derivative is 3x² * 2tan(sin(x))sec²(sin(x))
Step-by-step explanation:
To find the derivative of the function f(x) = x³ tan(sin(x))², we will use the chain rule. The chain rule states that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function.
The derivative of x³ is 3x², and the derivative of tan(sin(x))² can be found using the chain rule. Let's denote u = sin(x), so the function becomes f(x) = x³ tan(u)². The derivative of u is cos(x), and the derivative of tan(u) is sec²(u) = sec²(sin(x)).
Using the chain rule, the derivative of the outer function x³ is 3x², and the derivative of the inner function tan(sin(x))² is 2tan(sin(x))sec²(sin(x)). Therefore, the derivative of f(x) = x³ tan(sin(x))² is:
f'(x) = 3x² * 2tan(sin(x))sec²(sin(x))