Question

In Exercises \(13-24,\) find \(d y / d x .\) If you are unsure of your answer, use NDER to support your computation. $$y=\sqrt{\tan 5 x}$$

Short answer

The derivative of the function \(y = \sqrt{\tan 5x}\) is: \(dy/dx = (5\sec^2 5x) / (2\sqrt{\tan 5x})\).

Step-by-step solution

Identifying the outer and inner function

Given the function \(y = \sqrt{\tan 5x}\), we can identify the outer function as \(f(u) = \sqrt{u}\) and the inner function as \(g(x) = \tan 5x\).

Differentiating the outer function

The derivative of the outer function \(f(u) = \sqrt{u}\) w.r.t. to \(u\) is \(f'(u) = 1 / (2\sqrt{u})\).

Differentiating the inner function

The derivative of the inner function \(g(x) = \tan 5x\) w.r.t. \(x\) is \(g'(x) = 5\sec^2 5x\). The derivative of \(\tan x\) is \(\sec^2x\), so in this case where we have \(\tan 5x\), the derivative by the chain rule becomes \(5\sec^2 5x\).

Applying the Chain Rule

The Chain Rule states that the derivative of the composed function \(f(g(x))\) is \(f'(g(x)) * g'(x)\). Applying this formula using the derivatives calculated in steps 2 and 3, we get \(dy/dx = f'(g(x)) * g'(x) = (1 / (2\sqrt{\tan 5x})) * (5\sec^2 5x)\).