List of top Mathematics Questions on Differentiation asked in UP Board XII

(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?
If \( y = \frac{x^2 + 3x + 4}{e^x \cos x} \), find \( \frac{dy}{dx} \).
Differentiate \( y = x^x + (\cos x)^{\tan x} \) with respect to \( x \).

If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \) for \( -1<x<1 \), then prove that \[ \frac{dy}{dx} = -\frac{1}{(1 + x^2)^2}. \] 

Differentiate \( y = (\cos x)^{\sin x \).}

If \( \cos y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that:

\[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a} \]

Find the differential coefficient of \( \cos(\sin x^2) \) with respect to \( x \).
If \( x = a \cos^2 t, y = b \sin^2 t \), then find \( \frac{dy}{dx} \).
Differential coefficient of $\cos^{-1}(e^x)$ will be:
For any positive constant \( a \), evaluate \[ \frac{dy}{dx}, \text{ where } y = a \frac{t+1}{t} \text{ and } x = (t + 1)^{\alpha}. \]
If \[ x = a\left(\cos t + \log \tan \frac{t}{2}\right), y = a \sin t, \] find the value of $\dfrac{dy}{dx}$.
If $y = \sin^{-1}x$, then prove that $(1 - x^{2}) \dfrac{d^{2}y}{dx^{2}} - x \dfrac{dy}{dx} = 0$.
If $y = 4t$ and $x = \dfrac{4}{t}$, then the value of $\dfrac{dy}{dx}$ will be: