Final answer:
The chain rule is applicable when finding the derivative of tan (x³), where you multiply the derivative of the outer function, sec²(u), by the derivative of the inner function, 3x², resulting in sec²(x³) · 3x².
Step-by-step explanation:
Yes, the chain rule is applicable when finding the derivative of tan (x³). The chain rule is a formula to compute the derivative of a composition of functions.
In this case, we have two functions, tan(u), where u is an inner function, and u(x) = x³. To use the chain rule, we take the derivative of the outer function (with the inner function plugged into it) and multiply it by the derivative of the inner function.
The derivative of tan(u) with respect to u is sec²(u), and the derivative of u(x) = x³ with respect to x is 3x². Therefore, the derivative of tan (x³) with respect to x is sec²(x³) · 3x².
This is a straightforward application of the chain rule, similar to the power rule of differentiation where we use the sum of derivatives in case of additional terms added together. Remember, the chain rule is a critical tool when dealing with composite functions in calculus.