Use logarithmic differentiation to find the derivative of the given function.
$y=\tan^{x}x$

• A. $\left(\tan x\right)^{x}\cdot\ln\left(\tan x\right)\cdot\sec^2x$
• B. $\left(\tan x\right)^{x-1}\cdot\sec^2x$
• C. $\ln\left(\tan x\right)+\frac{x\sec^2x}{\tan x}$
• D. $\left(\ln\left(\tan x\right)+\frac{x\sec^2x}{\tan x}\right)\cdot\tan^{x}x$

Answer: (D)

Verified step by step guidance

1. Start by taking the natural logarithm of both sides of the equation y = (tan x)^x. This gives us ln(y) = ln((tan x)^x).
2. Use the property of logarithms that allows you to bring the exponent down: ln(y) = x * ln(tan x).
3. Differentiate both sides with respect to x. On the left side, use implicit differentiation: d/dx[ln(y)] = (1/y) * dy/dx. On the right side, apply the product rule to differentiate x * ln(tan x).
4. The product rule states that d/dx[u*v] = u'v + uv'. Here, u = x and v = ln(tan x). Differentiate u and v separately: u' = 1 and v' = (1/tan x) * sec^2(x) using the chain rule.
5. Substitute the derivatives back into the product rule: dy/dx = y * (ln(tan x) + x * (sec^2(x)/tan x)). Finally, replace y with (tan x)^x to express dy/dx in terms of x.