List of top Mathematics Questions on Logarithmic Differentiation asked in CBSE CLASS XII

If \[ (\sin x)^y=y^{\cos x}, \] then find \[ \frac{dy}{dx}. \]
Given that \( y = (\sin x)^x \cdot x^{\sin x} + a^x \), find \( \frac{dy}{dx} \).
Find the interval in which the function \( f(x) = x^4 - 4x^3 + 10 \) is strictly decreasing.
If \( x^y = e^x - y \), prove that \( \frac{dy}{dx} = \frac{\log x (1 + \log x)^2}{1 + \log x} \).
If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).
Express \( \tan^-1 \left( \frac\cos x1 - \sin x \right) \), where \( -\frac\pi2<x<\frac\pi2 \), in the simplest form.
For what value of \( k \), the function given below is continuous at \( x = 0 \)?
\[ f(x) = \begin{cases} \frac{\sqrt{4} + x - 2x}{x}, & x \neq 0 \\ k, & x = 0 \end{cases} \]
If \( x^y = e^{x-y} \), prove that \(\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^{2}}\).