List of top Mathematics Questions on Differential Calculus

If \( y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) \), then \(\frac{dy}{dx}\) at \(x = \frac{\sqrt{5}}{2}\) is equal to:

The value of the first derivative of \(f(x)=\left(3x^3+2x^2+x+1\right)^2\) evaluated at \(x=2\) is (in integer).

Let \(f:\mathbb{R}\to\mathbb{R}\) be the function defined by \(f(x)=3e^x\cos 2x\). If \(p(x)=ax^3+bx^2+cx+d\) is the third degree Taylor polynomial of \(f\) at \(x=0\), then \(|a|+|b|+|c|+|d|\) equals (in integer).

Let \(f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\) and \(g:\mathbb{R}\setminus\{0\}\to \mathbb{R}\) be two bounded and differentiable functions. Define \(F:\mathbb{R}^2\to \mathbb{R}\) by

Let \(h:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be two bounded functions. Define the function \(f:\mathbb{R}\to\mathbb{R}\) by

Let \(g:\mathbb{R}\to\mathbb{R}\) be a function defined by

Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be such that $f(e)=\frac{7}{4}$ and $f^{\prime}(x^{2})=\frac{3\log x}{x^{2}}$, then $f(e^{2})=$

The value of \(c\) of Lagrange's Mean Value Theorem for \[ f(x)=\sqrt{25-x^2} \] on \([1,5]\) is:

If $y = \sqrt{\tan x + y}$, then $\frac{dy}{dx} = $

The second order derivative of $\sec^{-1}\left(\frac{1}{2x^2 - 1}\right)$ with respect to $\cos^{-1}(2x^2 - 1)$, where $0<x<\frac{1}{\sqrt{2}}$ is

If $\sqrt{x} \sqrt[3]{y} = (x + y)^n$ and $x\frac{dy}{dx} - y = 0$, then $n =$

The point of inflection of the function $f(x)=x^3$ in the interval $[-1,1]$ is

If \( y = f(x) \) satisfies the differential equation \[ (x^2 - 4)y' - 2xy + 2x(4 - x^2)^2 = 0 \] and \( f(3) = 15 \), then find the local maximum value of \( f(x) \):

The left hand derivative of $|x|$ with respect to $x$ at $x=0$ is

If $m$ and $n$ are respectively the order and degree of the differential equation $2x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+y=0$, then $m+n=$

Statement I: The function $f(x)=x^2$ is decreasing in the interval $(0,\infty)$.
Statement II: Any function $y=f(x)$ is decreasing if $\dfrac{dy}{dx}& Lt;0$.
Which of the following is correct?

The angle $\theta$, at which the curves $y = 3^x$ and $y = 7^x$ intersect, is given by ______.

If the straight line \( \frac{x}{p} - \frac{y}{q} = 1 \) touches the curve \( \left( \frac{x}{a} \right)^n - \left( \frac{y}{b} \right)^n = 1 \), then

If $f(x)=3x^{2}+2xf^{\prime}(1)+f^{\prime\prime}(2)$, then $f(x)=..........$

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a>0 \), attains its local maximum and minimum at \( p \) and \( q \), respectively, such that \( p^2 = q \), then \( f(3) \) is equal to: