List of top Mathematics Questions on Differentiation asked in CBSE Class Twelve Board Exam

If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).
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: \[ \frac{dy}{dx}, \quad \text{if} \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]
Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad \text{w.r.t.} \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]
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 \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), then prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
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 $\tan^{-1}(x^2 + y^2) = a^2$, then find $\frac{dy}{dx}$.
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 \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \), for \( -1<x<1, x \neq y \), then prove that \[ \frac{dy}{dx} = \frac{-1}{(1 + x)^2}. \]

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

A small town is analyzing the pattern of a new street light installation. The lights are set up such that the intensity of light at any point \( x \) metres from the start of the street can be modeled by:\[ f(x) = e^x \sin x \] where \( x \) is in metres.Based on this, answer the following:
A small town is analyzing the pattern of a new street light installation.
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 \( b \).
Find the intervals in which the function \( f(x) = \frac{\log x}{x} \) is strictly increasing or strictly decreasing.
Find the value of \( a \) and \( b \) so that the function \( f(x) \), defined as: \[ f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & x < 2, \\ a + b, & x = 2, \\ \frac{x - 2}{|x - 2|} + b, & x > 2, \end{cases} \] is a continuous function.
If \( x \cos(p + y) + \cos p \sin(p + y) = 0 \), prove that \( \cos p \frac{dy}{dx} = -\cos^2(p + y) \), where \( p \) is a constant.
If \( x = e^{x/y} \), prove that \( \frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2} \).
\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:

Of the following, which group of constraints represents the feasible region given below? 

If \( y = \sin^{-1} x \), then \( \frac{d^2 y}{dx^2} \) is:
Show that \( f(x) = \frac{4 \sin x}{2 + \cos x} - x \) is an increasing function of \( x \) in \( \left[ 0, \frac{\pi}{2} \right] \).
If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).
Given that \( x^y + y^x = a^b \), where \( a \) and \( b \) are positive constants, find \( \frac{dy}{dx} \).
If \( e^{x^2y} = c \), then \( \frac{dy}{dx} \) is: