Question

Find the first derivative of y = tan(1/x).
Suppose that f'(4) = g(4) = g'(4) = 1.

Answer 1

The first derivative of y = tan(1/x) is -sec^2(1/x)/x^2.

Step-by-step explanation:

The first derivative of y = tan(1/x) can be found using the chain rule. Let's define u = 1/x. Then, y = tan(u). Applying the chain rule, we have:

dy/dx = (dy/du) * (du/dx)

To find dy/du, we differentiate y = tan(u) with respect to u using the derivative of the tangent function, which is sec^2(u):

dy/du = sec^2(u)

To find du/dx, we differentiate u = 1/x with respect to x using the power rule:

du/dx = -1/x^2

Substituting these results back into the chain rule equation, we have:

dy/dx = sec^2(u) * (-1/x^2) = -sec^2(u)/x^2

So, the first derivative of y = tan(1/x) is -sec^2(1/x)/x^2.