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

If $x=\sin t-\cos t$ and $y=\sin t\cos t$, find $\dfrac{dy}{dx}$ at $t=\dfrac{\pi}{4}$.
If $e^{x+y}+y=3x$, then $\dfrac{dy}{dx}$ is
The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate is its area increasing when the side of the triangle is 15 cm?
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), then prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad \text{w.r.t.} \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]
(a) Differentiate $ \left( \frac{5x}{x^5} \right)$ with respect to $x$.
If $\tan^{-1}(x^2 + y^2) = a^2$, then find $\frac{dy}{dx}$.

A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
On the basis of the above information, answer the following questions :
Find \( \frac{dS}{dx} \).

Differentiate \( \log(x^2 + \csc^2 x) \) with respect to \( x \).

Find the interval in which $f(x) = x + \frac{1}{x}$ is always increasing, $x \neq 0$.

If $f(x) = x^x$, find the critical point:
Differentiate \( y = \sqrt{\log \sin \left( \frac{x^3}{3} - 1 \right)} \) with respect to \( x \).
(b) If $2x^2 - 5xy + y^3 = 76$, then find $\frac{dy}{dx}$.
Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) with respect to x.
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) with respect to x.
For a positive constant \( a \), differentiate \( \left( t + \frac{1}{t} \right)^a \) with respect to \( t \), where \( t \) is a non-zero real number.
Check the differentiability of the function f(x) = |x| at x = 0.
Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]
Find: \[ \frac{dy}{dx}, \quad \text{if} \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]
The area of an expanding rectangle is increasing at the rate of \( 4 \, \text{cm}^2/\text{s} \). The length of the rectangle is always square of its breadth. At what rate is the length of the rectangle increasing at an instant, when breadth \( = 4.5 \, \text{cm} \)?
If \( \int_a^b x^3 dx = 0 \) and \( \int_a^b x^2 dx = \frac{2}{3} \), then find the values of \( a \) and \( b \).
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \), for \( -1<x<1, x \neq y \), then prove that \[ \frac{dy}{dx} = \frac{-1}{(1 + x)^2}. \]
If \( y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 \), then show that \( x(x + 1)^2 y_2 + (x + 1)^2 y_1 = 2 \).

If \( f(x) = \begin{cases} 2x - 3, & -3 \leq x \leq -2 \\x + 1, & -2<x \leq 0 \end{cases} \), check the differentiability of \( f(x) \) at \( x = -2 \).