Question

If, \(y = x^{\tan(x)}\), then \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\), is

• A. \(\frac{\pi}{4}\)
• B. \(\frac{\pi}{4}\log(\frac{\pi}{4})\)
• C. \(\frac{\pi}{4}(\log(\frac{\pi}{4}))^2+1\)
• D. \(\frac{\pi}{4}\log(\frac{\pi}{4}) + 2\log(\frac{\pi}{4})\)
• E. None of the above.

Hint:

Logarithmic differentiation is essential for functions of the form \(f(x)^{g(x)}\). Remember the process: take logs, differentiate using product/chain rules, solve for \(y'\), and substitute back the original function for \(y\). Double-check your differentiation rules and algebraic manipulations, as errors are common.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The function is of the form \(y = f(x)^{g(x)}\). To differentiate such functions, we use logarithmic differentiation. This involves taking the natural logarithm of both sides, differentiating implicitly, and then solving for \(\frac{dy}{dx}\).

Step 2: Key Formula or Approach:
1. Let \(y = x^{\tan(x)}\).
2. Take the natural logarithm: \(\ln(y) = \ln(x^{\tan(x)}) = \tan(x) \ln(x)\).
3. Differentiate both sides with respect to \(x\) using the product rule.
4. Solve for \(\frac{dy}{dx}\) and substitute \(x = \frac{\pi}{4}\).

Step 3: Detailed Explanation:
We have \(y = x^{\tan(x)}\).
Taking the natural logarithm on both sides: \[ \ln(y) = \tan(x) \ln(x) \] Differentiating with respect to \(x\): \[ \frac{d}{dx}(\ln(y)) = \frac{d}{dx}(\tan(x) \ln(x)) \] Using the chain rule on the left and the product rule on the right: \[ \frac{1}{y} \frac{dy}{dx} = \left(\frac{d}{dx}\tan(x)\right)\ln(x) + \tan(x)\left(\frac{d}{dx}\ln(x)\right) \] \[ \frac{1}{y} \frac{dy}{dx} = \sec^2(x)\ln(x) + \tan(x) . \frac{1}{x} \] Solving for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = y \left( \sec^2(x)\ln(x) + \frac{\tan(x)}{x} \right) \] Substitute \(y = x^{\tan(x)}\): \[ \frac{dy}{dx} = x^{\tan(x)} \left( \sec^2(x)\ln(x) + \frac{\tan(x)}{x} \right) \] Now, we evaluate this derivative at \(x = \frac{\pi}{4}\).
At \(x = \frac{\pi}{4}\):

\(\tan(\frac{\pi}{4}) = 1\)
\(\sec(\frac{\pi}{4}) = \sqrt{2} \implies \sec^2(\frac{\pi}{4}) = 2\)
\(y(\frac{\pi}{4}) = (\frac{\pi}{4})^{\tan(\frac{\pi}{4})} = (\frac{\pi}{4})^1 = \frac{\pi}{4}\) \end{itemize} Substituting these values into the derivative expression: \[ \frac{dy}{dx}\bigg|_{x=\pi/4} = \frac{\pi}{4} \left( 2 . \ln(\frac{\pi}{4}) + \frac{1}{\pi/4} \right) \] \[ = \frac{\pi}{4} \left( 2\ln(\frac{\pi}{4}) + \frac{4}{\pi} \right) \] \[ = \frac{\pi}{4} . 2\ln(\frac{\pi}{4}) + \frac{\pi}{4} . \frac{4}{\pi} \] \[ = \frac{\pi}{2}\ln(\frac{\pi}{4}) + 1 \]
Step 4: Final Answer:
The calculated value of the derivative at \(x = \frac{\pi}{4}\) is \(\frac{\pi}{2}\ln(\frac{\pi}{4}) + 1\). This result does not match any of the options (A), (B), (C), or (D). Therefore, the correct choice is "None of the above". The provided options appear to be incorrect.