Question

If f is a differentiable function and y=sin(f(x²)), what is dy/dx​ when x=3?

Answer 1

To find dy/dx, apply the chain rule by differentiating the outer function and the inner function separately, then multiply the derivatives. Substitute x = 3 to find the value of dy/dx.

Step-by-step explanation:

To find dy/dx, we need to apply the chain rule. Let's break down the steps:

1. First, differentiate the outer function y = sin(f(x^2)) with respect to the inner function f(x^2). The derivative of sin(f(x^2)) is cos(f(x^2)) * f'(x^2).
2. Next, differentiate the inner function f(x^2). The derivative of f(x^2) with respect to x is 2x * f'(x^2).
3. Finally, multiply the derivatives you obtained in step 1 and step 2 to find the derivative of the whole function y = sin(f(x^2)) with respect to x.

Substituting x = 3 into the derivative(dy/dx) will give you the value. Plug in x = 3, f'(x^2) = f'(9), and evaluate the expression to find the exact value of dy/dx at x = 3.