Question

Differentiate the function f(x)=tan⁻¹ (2x) using exact values.

Answer 1

The derivative of the function f(x) = tan⁻¹(2x) using exact values is dy/dx = 2/(1 + 4x²).

Step-by-step explanation:

To differentiate the function f(x) = tan⁻¹(2x) using exact values, we'll apply the chain rule. Let u = 2x, and y = tan⁻¹(u). The derivative of y with respect to x is given by:

dy/dx = dy/du * du/dx

To find dy/du, we use the derivative of arctangent, which is 1/(1 + u²). To find du/dx, we differentiate u with respect to x, giving 2. Substituting these values into the chain rule formula, we have:

dy/dx = 1/(1 + (2x)²) * 2

Expanding and simplifying, the derivative is:

dy/dx = 2/(1 + 4x²)