Question

Find the derivative of the function. y=(tan⁻¹ (4x))²

Answer 1

To find the derivative of the function y=(tan⁻¹ (4x))², use the chain rule by differentiating the outer and inner functions separately and then multiplying the results.

Step-by-step explanation:

To find the derivative of the function y=(tan⁻¹ (4x))², we can apply the chain rule. Let's break it down:

1. Begin by differentiating the outer function, which is y². Using the power rule, the derivative is 2y.
2. Next, differentiate the inner function, which is tan⁻¹ (4x). The derivative of tan⁻¹ u is 1/(1+u²), and the derivative of 4x is 4.
3. Now, multiply the derivative of the outer function (2y) with the derivative of the inner function (1/(1+u²) * 4).

This gives us the derivative of y in terms of x: dy/dx = 2y * 4/(1+(4x)²).