CUET Mathematics: CalculusContinuity & Differentiability. Free, no login required.

If $x = a\left(t - \dfrac{1}{t}\right)$, $y = b\left(t + \dfrac{1}{t}\right)$, then $\dfrac{dy}{dx} =$

If $y = \left[\dfrac{2}{1+x^{b-a}+x^{c-a}} + \dfrac{2}{1+x^{c-b}+x^{a-b}} + \dfrac{2}{1+x^{a-c}+x^{b-c}}\right]^x$, then $\dfrac{dy}{dx}$ is equal to :

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$, then the value of $f(0)$ is: (Take $\log_e x = \log x$)

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If $x = at^2$ and $y = 4at$, then the value of $\frac{d^2y}{dx^2}$ at $t = 1/a$, is

If $f(x) = \begin{cases}\lambda x+5, & if\ x\le 2\\x+3, & if\ x>2\end{cases}$ is continuous at $x=2$, then, the value of $\lambda$ is

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If $y = \log_3(\log x)$, (Consider $\log_e x = \log x$) then the value of $\frac{dy}{dx}$ is

If $y=e^{2\log_e t}$ and $x=\log_e e^{t^2}$, then the value of $\frac{d^2y}{dx^2}$ is

Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$. Then, the value of $\lambda$ is equal to:

If the function $f(x)=\begin{cases}\frac{x^3-1}{x-1}, & x<1\\a, & x=1\\\frac{b\sin(x-1)}{x-1}, & x>1\end{cases}$ is continuous at $x=1$, then which of the following statements are correct ?

A. $a=b$

B. $a+b$ is a multiple of 4

C. $2a-b$ is a multiple of 3

D. $a+4b$ is divisible by 5

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If $x\sqrt{1+y}+y\sqrt{1+x}=0, x>-1$ and $x\neq y$, then $(1+x)^2\frac{dy}{dx}$ is equal to

if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

Let $x=at^2, y=2at$. If $\frac{d^2y}{dx^2}+\frac{dy}{dx}=0$ at $t=3$, then $a$ is equal to:

If $y = \log(\log x)$, then $\frac{d^2y}{dx^2}$ is equal to: (Consider $\log_e x = \log x$)

If $x=a(1-\cos\theta), y=a(\theta+\sin\theta)$ then $\frac{d^2y}{dx^2}$ is equal to

The function $f(x)=\begin{cases}\frac{e^{3x}-e^{-5x}}{x}, & x\neq 0\\ \lambda, & x=0\end{cases}$ is continuous at x=0 for the value of '$\lambda$' as:

A function f(x) is defined by $f(x)=\begin{cases}x^2+\alpha x+\beta: & 0\leq x<2\\3x+2 : & 2\leq x\leq4\\2\alpha x+5\beta :& 4<x\leq8\end{cases}$. If function f(x) is continuous on [0, 8], then value of $2\alpha+3\beta$ is

If $x=4t^3$ and $y=6t^2$ then $\frac{d^2y}{dx^2}$ at $t=1$ is

If $e^y(1+x)=1$, then $\dfrac{d^2y}{dx^2}$ is:

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If the function $f(x)=\begin{cases}3x-6, & x \le 5 \\ 4k, & x>5\end{cases}$ is continuous at $x=5$, then the value of $k$ is:

For the function $f(x)=|\cos x|$, which of the following statements are correct?

(A) $f(x)$ is continuous for all $x \in \mathbb{R}$

(B) $f(x)$ is differentiable for all $x \in \mathbb{R}$

(C) $f(x)$ is not differentiable at $x=(2n+1)\dfrac{\pi}{2}, n \in \mathbb{Z}$

(D) $f(x)$ is not differentiable at $x=n\pi, n \in \mathbb{Z}$

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If $e^y(x+1)=1$, then

If $xy\log_e(x+y) = 1$, then $\dfrac{dy}{dx} =$

If $y = (\log_e x)^{\log_e x}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(x) = |3-x| + (3+x)$ be a function, then which of the following statements are TRUE?

A. $f(x)$ is continuous at $x=3$

B. $f(x)$ is continuous at $x=-3$

C. $f(x)$ is differentiable at $x=3$

D. $f(x)$ is differentiable at $x=-3$

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If $y = ax^{n+2} + \dfrac{b}{x^{n+1}}$, then $x^2\dfrac{d^2y}{dx^2}$ is equal to

If $y = x^3\log(3x^2+6)$, then $\dfrac{dy}{dx} =$