Derivative of $\sec^{-1} \left( \frac{1}{2x^2 + 1 } \right)$ w.r.t. $\sqrt{ 1 + 3x} $ at $x = - \frac{1}{3}$ is

Let $y = \sec^{-1} \left(\frac{1}{2x^{2}+1}\right) $ and $z=\sqrt{1+3x}$ $ \therefore \frac{dy}{dz} = \frac{dy/dx}{dz/dz}$ $ = \left(2x^{2} +1\right) . \frac{1}{\sqrt{\left(\frac{1}{2x^{2} +1}\right)^{2} -1} }$ $ . \frac{-4x}{ \left(2x^{2}+1\right)^{2}} / \frac{1}{2} \left(1+3x\right)^{-1/2}.3$ $ = \frac{-4x}{\sqrt{1-\left(2x^{2}+1\right)^{2}}} . \frac{2}{3} \sqrt{1-3x}$ At $x = \frac{1}{3}, \frac{dy}{dz} = 0 $

Derivative of $\sec^{-1} \left( \frac{1}{2x^2 + 1

Top Questions on limits and derivatives

Let $\left\lfloor t \right\rfloor$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which

\[ \lim_{x \to 0^+} \left( x \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \dots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \dots + \left\lfloor \frac{9^2}{x^2} \right\rfloor \right) \geq 1 \]

is equal to __________.
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Evaluate the following limit: $ \lim_{n \to \infty} \prod_{r=3}^n \frac{r^3 - 8}{r^3 + 8} $.
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If $ f(x) $ is defined as follows:
$$ f(x) = \begin{cases} 4, & \text{if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & \text{if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & \text{if } \sqrt{5} \leq x < \infty. \end{cases} $$ If $ k $ is the number of points where $ f(x) $ is not differentiable, then $ k - 2 = $
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Let $ f $ be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:
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Let the function $ g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $ be given by $ g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} $. Determine the properties of $ g $.
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Notes on limits and derivatives

Concepts Used:

Limits And Derivatives

Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.

Limits Formula:

Derivatives of a Function:

A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.

Properties of Derivatives:

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