Question:

<b>Implicit Differentiation</b><br>Differentiate both sides w.r.t. x:<br>$\log_a(\log_a x) + \frac{x}{\ln a \cdot \log_a x} \cdot \frac{1}{x\ln a} - 2x + 2y\frac{dy}{dx} = 0$<br>At $x=e$: $\log_a(\log_a e) = \log_a\\!\left(\frac{1}{\ln a}\right)$; the simplification gives $2y\frac{dy}{dx} = 2e - \frac{1}{e\,(\ln a)^2}\cdot\frac{1}{\log_a e}$. Using the implicit equation at $x=e$: $y^2 = 4+e^2$, so $y=\sqrt{4+e^2}$. Differentiating cleanly: $\frac{dy}{dx} = \frac{2e-1}{2\sqrt{4+e^2}}$.<br><b>Key concept:</b> Implicit differentiation + logarithmic identity $\log_a x = \frac{\ln x}{\ln a}$.<br><b>Trap:</b> Don't forget to differentiate the $x\\!\cdot\\!\log_a(\log_a x)$ term using the product rule.