Question

If $y=(\tan x)^{x}$, then $\frac{1}{y}\frac{dy}{dx}=$ ________.

• A. $\log(\tan x)+2x \csc(2x)$
• B. $\log(\tan x)+x \csc(2x)$
• C. $x \log(\tan x)+2x \csc(2x)$
• D. $x \log(\tan x)+x^{2} \csc(2x)$
• E. $\log(\tan x)+\frac{x}{2} \csc(2x)$

Hint:

Use log differentiation for functions of the form $f(x)^{g(x)}$.

Answer 1

The Correct Option is A

Step 1: Concept
Use logarithmic differentiation: $\ln y = x \ln(\tan x)$.

Step 2: Meaning
Differentiate: $\frac{1}{y}\frac{dy}{dx} = 1 \cdot \ln(\tan x) + x \cdot \frac{1}{\tan x} \cdot \sec^2 x$.

Step 3: Analysis
Simplify: $\frac{\sec^2 x}{\tan x} = \frac{1}{\cos^2 x} \cdot \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x} = 2 \csc 2x$.

Step 4: Conclusion
$\frac{1}{y}\frac{dy}{dx} = \ln(\tan x) + 2x \csc 2x$. Final Answer: (A)