List of practice Questions

Let \(f(x) = \frac{\sin x}{x}\) for \(x \neq 0\). Then the value of \(f'(\frac{\pi}{2})\) is equal to
If \(y = \sin\left(\tan^{-1}\left(\frac{1}{\sqrt{x^2 - 1}}\right)\right), x > 1\), then \(\frac{dy}{dx} =\)
If \(y = \sqrt{x} + 2\cos(\sqrt{x})\), then the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi^2}{4}\) is equal to
Let $f(x) = \frac{1 + \tan^2 x}{1 - \tan^2 x}$ for $0 < x < \frac{\pi}{4}$. Then the value of $f'(\frac{\pi}{8})$ is equal to
The point $P(x, y)$, where $y = 4\log_e(2)$, lies on the curve with equation $y = \log_e(x^3 + 24)$. Then the value of $\frac{dy}{dx}$ at the point $P$ is
A cubic curve $y = f(x)$ passes through the points $(1, -7)$ and $(2, 11)$. If $\frac{dy}{dx} = 6x^2 + kx - 5$, where $k$ is a constant, then $f(x) =$}
If $f^{\prime}(5)=\frac{3}{5}$ then the value of $\lim_{h\rightarrow 0}\frac{f(5+10h)-f(5)}{h}$ is equal to
If $f(x)=\frac{|x|}{x^{2}}$ then $f^{\prime}(2)$ is equal to
If $f(x)=\tan^{-1}\left(\frac{3\cos x-5\sin x}{5\cos x+3\sin x}\right)$ , then the value $f^{\prime}(1)$ is
If \( h(x) = \sqrt{4f(x) + 3g(x)} \), \( f(1)=4 \), \( g(1)=3 \), \( f'(1)=3 \), \( g'(1)=4 \), then \( h'(1) \) is equal to:}
Find the derivative of tan⁻¹(x) + cos⁻¹(x)
If $e^{y}+x^{2}y+xy^{2}=e^{1}$, then $\frac{dy}{dx}$ at (0,1) is equal to ________.
If $f(x)=x|x|$, then $f^{\prime}(-10)=$ ________.
If $y = \tan^{-1} \left[ \frac{12x - 64x^3}{1 - 48x^2} \right]$, then $dy/dx = \dots$
If $f(x) = \sqrt{1 + \cos^2(x^2)}$, then $f'(\frac{\sqrt{\pi}}{2})$ is ______.
If $f(x) = \frac{\sin^2 x}{1+\cot x} + \frac{\cos^2 x}{1+\tan x}$, then the value of $f'(\frac{\pi}{6})$ is equal to ______.
If $y = \tan^{-1} \left( \sqrt{\frac{1+\sin x}{1-\sin x}} \right)$, $0 \le x < \frac{\pi}{2}$, then $y' \left( \frac{\pi}{6} \right) = $ ______.
If $y=\tan^{-1}(\frac{1}{1+x+x^{2}})+\tan^{-1}(\frac{1}{x^{2}+3x+3})+\tan^{-1}(\frac{1}{x^{2}+5x+7})$ then $y^{\prime}(0)$ is}
If $u=\log(\sqrt{x+1}-\sqrt{x-1})$ and $v=\sqrt{x+1}+\sqrt{x-1}$ then $\frac{du}{dv}=...$
If $u=\log(\sqrt{x+1}-\sqrt{x-1})$ and $v=\sqrt{x+1}+\sqrt{x-1}$ then $\frac{du}{dv}=...$
If $$ y = \sin^{-1} \left( \frac{2x}{1+x^2} \right) + \sec^{-1} \left( \frac{1+x^2}{1-x^2} \right) $$ then the value of $$ \frac{dy}{dx} $$ at $$ x = \sqrt{3} $$ is
Let \(( f(x) =\) \(\int_0^{x^2 \frac{t^2 - 8t + 15}{e^t} dt}\), \(x \in \mathbb{R}\). Then the numbers of local maximum and local minimum points of \( f \), respectively, are:
Given \( f'(1) = 3 \), \( f(1) = 1 \), and
\[ y = f\left(f(f(x))\right) + \left(f(x)\right)^2, \] then find \( \frac{dy}{dx} \) at \( x = 1 \).
If $ y = \frac{b}{a} $, then $ \frac{dy}{dx} $ is:
Find the values of \( a \) for which \( f(x) = \sqrt{3} \sin x - \cos x - 2ax + b \) is decreasing on \( \mathbb{R} \).