Differentiation practice · Mixed difficulty

Let $y=y(x)$ be $a$ function of $x$ satisfying $y \sqrt{1-x^{2}}=k-x \sqrt{1-y^{2}}$ where $k$ is $a$ constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{d y}{d x}$ at $x=\frac{1}{2}$, is equal to:

If $x=f(t), y=g(t)$, then $\frac{d^2 y}{d x^2}=\ldots \ldots$

$f\left( x \right)$ is a polynomial of degree 2, $f\left( 0 \right) = 4,f'\left( 0 \right) = 3$ and $f''\left( 0 \right) = 4$, then $f\left( { - 1} \right)$ is equal to

If $2{x^2} - 3xy + {y^2} + x + 2y - 8 = 0,$ then $\frac{{dy}}{{dx}}$ is equal to

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be differentiable functions such that $f(x)=x^3+3 x+2, g(f(x))=x$ for all $x \in R$. Then, $g^{\prime}(2)=$

If $f(x)=\cos x \cos 2 x \cos 2^2 x \cos 2^3 x \ldots \ldots$ $\cos 2^{n-1} x$ and $n>1$, then $f^{\prime}\left(\frac{\pi}{2}\right)$, is

The derivative of $y=x^{\ln x}$ is

On the curve $y=x^3$, the point at which the tangent line in parallel to the chord through the point $(-1,-1)$ and $(2,8)$ is

If $f(x)=(1+x)^n$, then the value of $f(0)+$ $f^{\prime}(0)+\frac{1}{2 !} f^{\prime \prime}(0)+\cdots+\frac{1}{n !} f^n(0)$ is

If $f(x)=x^{n}$, then the value of
$f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\ldots \ldots \ldots . . \frac{(-1)^{n} f^{n}(1)}{n !}$ is

Let $\mathrm{f}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be two functions defined by

Statement I: $f$ is a continuous function at $\mathrm{x}=0$.
Statement II: $g$ is a differentiable function at $\mathrm{x}=0$.

Let $f(\theta)=\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$. Then, the value of $\frac{d}{d(\tan \theta)} \cdot f(\theta)$, is

If $\cos (x+y)=y \sin x$, then $\frac{d y}{d x}$ is equal to

$\frac{d}{d x}\left[e^{-a x^{2}} \log (\sin x)\right]$

If ${x^2} + {y^2} = t + \frac{1}{t}$ and ${x^4} + {\mathrm{y}^4} = {\mathrm{t}^2} + \frac{1}{{{\mathrm{t}^2}}},$ then $\frac{{dy}}{{dx}}$ is equal to

If $y = ({\log _{\cos x}}\sin x)({\log _{\sin x}}\cos x) + {\sin ^{ - 1}}\frac{{2x}}{{1 + {x^2}}},$ then $\frac{{\mathrm{d}y}}{{\mathrm{d}x}}$ at $x = \frac{\pi }{2}$ is equal to

The function $f(x)=e^x+x$, being differentiable and one-one, has a differentiable inverse $f^{-1}(x)$. The value of $\frac{d}{d x}\left(f^{-1}\right)$ at the point $f(\log 2)$ is

A particle is moving along the parabola $y^2=4(x+2)$. As it passes through the point $(7,6)$ its $y$-coordinate is increasing at the rate of 3 units per second. The rate at which $x$-coordinate change at this instant is (in units/sec)

The derivative of $f(x)=3|2+x|$ at the point $x_{0}=-3$ is

Let $f$ be a bijection satisfying $f^{\prime}(x)=f(x)$. Then, $\left(f^{-1}\right)^{\prime \prime}(x)$ is equal to

Let $f: R \rightarrow R$ be a function given by $f(x+y)=f(x) f(y)$ for all $x, y \in R$

If $f(x) \neq 0$ for all $x \in R$ and $f^{\prime}(0)$ exists, then $f^{\prime}(x)$ equals

If $f(x+y)=f(x) \cdot f(y) \forall x \cdot y$ and $f(5)=2$, $f^{\prime}(0)=3$, then $f^{\prime}(5)$ is

If $x^{2}+y^{2}+\sin y=4$, then the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,0)$ is

If $f(x)=\log |2 x|, x \neq 0$ then $f^{\prime}(x)$ is equal to

$\frac{d}{d x} \cos ^{-1} \frac{x-x^{-1}}{x+x^{-1}}$ is equal to

If $f\left( x \right) = {x^n},$ then the value of $f\left( 1 \right) - \frac{{f'\left( 1 \right)}}{{1!}} + \frac{{f''\left( 1 \right)}}{{2!}} - \frac{{f'''\left( 1 \right)}}{{3!}} + \ldots + \frac{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)}}{{n!}}$ is

Let $f$ be a twice differentiable function such that $f''\left( x \right) = - f\left( x \right)$ and $f'\left( x \right) = \mathrm{g}\left( x \right).$ If $h'\left( x \right) = \left[ {f{{\left( x \right)}^2} + \mathrm{g}{{\left( x \right)}^2}} \right]h\left( 1 \right) = 8$ and $h\left( 0 \right) = 2,$ then $h\left( 2 \right)$ is equal to

Which of the following is/are true?

I. The derivative of $\sin (x+a)$ is $\cos (x+a)$, where $a$ is a fixed non-zero constant.

II. The derivative of $\operatorname{cosec} x \cot x$ is $\operatorname{cosec}^3 x-\cot ^2 x \operatorname{cosec} x$

If $f''\left( x \right) = - f\left( x \right)$, where $f\left( x \right)$ is a continuous double differentiable function and $\mathrm{g}\left( x \right) = f'\left( x \right).\,$ If $F\left( x \right) = {\left( {f\left( {\frac{x}{2}} \right)} \right)^2} + {\left( {\mathrm{g}\left( {\frac{x}{2}} \right)} \right)^2}$ and $F\left( 5 \right) = 5$ then $F\left( {10} \right)$

If $u = \log \left( {\frac{{{x^2} + {y^2}}}{{x + y}}} \right),$ then the value of $x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}$ is