Question

Find the derivative of the function y = sin(tan 9x)?

Answer 1

The derivative of the function y = sin(tan(9x)) is found using the chain rule and results in dy/dx = 9 * cos(tan(9x)) * sec^2(9x).

Step-by-step explanation:

The student was asked to find the derivative of the function y = sin(tan(9x)). To do this, we must apply the chain rule for differentiation which involves taking the derivative of the outer function and multiplying it by the derivative of the inner function. Here's how you can do it step by step:

1. First, take the derivative of the outer function, which is sin, with respect to its argument. The derivative of sin(u) is cos(u), where u is any differentiable function.
2. Next, differentiate the inner function, which is tan(9x), with respect to x. The derivative of tan(v) is sec^2(v) where v is a function of x, and in this case, v is 9x. The derivative of 9x with respect to x is 9.
3. Thus, by the chain rule, the derivative of y with respect to x is cos(tan(9x)) multiplied by sec^2(9x) multiplied by 9.

Therefore, dy/dx = 9 * cos(tan(9x)) * sec^2(9x).