Differentiation

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 $\frac{d y}{d x}$ at $x=\frac{\sqrt{3}}{2}$ is equal to :

Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$, $x \in \mathbb{R}$. Then the value of $f'(5)$ is :

$$ \text { If } y(x)=\left|\begin{array}{ccc} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array}\right|, x \in \mathbb{R} \text {, then } \frac{d^2 y}{d x^2}+y \text { is equal to } $$

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(x) \geq 0, f^{\prime\prime}(x)=f(x), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _e 3\right)$ is equal to __________ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all $x, y \in \mathbf{R}$.
If $f^{\prime}(0)=\frac{1}{2}$, then the...

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to :

Let \(f(x)=a x^3+b x^2+c x+41\) be such that \(f(1)=40, f^{\prime}(1)=2\) and \(f^{\prime \prime}(1)=4\). Then \(a^2+b^2+c^2\) is equal to:

If \(\log _e y=3 \sin ^{-1} x\), then \((1-x^2) y^{\prime \prime}-x y^{\prime}\) at \(x=\frac{1}{2}\) is equal to

Let \(f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}(1)=\lim _\limits{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)\). Then $$\lim _\limits{a \rightarrow \infty} \frac{a(a+1)}{2}...

$$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$$

Suppose for a differentiable function \(h, h(0)=0, h(1)=1\) and \(h^{\prime}(0)=h^{\prime}(1)=2\). If \(g(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}\), then \(g^{\prime}(0)\) is equal to:

If \(y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}\), then at \(\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y\) is equal to :

Let \(f(x)=x^5+2 \mathrm{e}^{x / 4}\) for all \(x \in \mathbf{R}\). Consider a function \(g(x)\) such that \((g \circ f)(x)=x\) for all \(x \in \mathbf{R}\). Then the value of \(8 g^{\prime}(2)\) is :

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of $$\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \...

If $$f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x \end{array}\right|,$$ then \(\frac{1}{5} f^{\prime}(0)=\) is equal to :

Let \(g: \mathbf{R} \rightarrow \mathbf{R}\) be a non constant twice differentiable function such that \(\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)\). If a real valued function \(f\) is defined a...

Let \(f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) be a function satisfying \(f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}\) for all \(x, y, f(y) \neq 0\). If \(f^{\prime}(1)=2024\), then

Suppose \(f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}\). Then the value of \(f^{\prime}(0)\) is equal to

\(\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }\)

Let $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then $f^{\prime}(10)$ is equal to ____________.

If $y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x$, then $96 y^{\prime}\left(\frac{\pi}{6}\right)$ is equal to :

Let \(f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}\). Then \(f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)\) is equal to

If \(2 x^{y}+3 y^{x}=20\), then \(\frac{d y}{d x}\) at \((2,2)\) is equal to :

Let \(y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)\). Then, at x = 1,

Let \(f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}\) For \(\mathrm{n} \geq 2\), define \(f^{\mathrm{n}}(x)=f^{1} \mathrm{o} f^{\mathrm{n}-1}(x)\). If $$f^{5}(x)=\frac{\mathrm{a} x+\mathrm{b}}{\mathrm{b} x+\math...

Let \(f:\mathbb{R}\to\mathbb{R}\) be a differentiable function that satisfies the relation \(f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}\). If \(f'(0)=2\), then \(|f(-2)|\) is equal to ___________.

Let \(f\) and \(g\) be the twice differentiable functions on \(\mathbb{R}\) such that
\(f''(x)=g''(x)+6x\)
\(f'(1)=4g'(1)-3=9\)
\(f(2)=3g(2)=12\).
Then which of the following is NOT true?

Let \(y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})\). Then \(y' - y''\) at \(x = - 1\) is equal to

If \(f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}\), then

If \(f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)\) and \(g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)\), then the value of \(f(4)-g(4)\) is equal to ____________.

Let \(f(x) = 2x + {\tan ^{ - 1}}x\) and \(g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]\). Then

If \(y(x)=x^{x},x > 0\), then \(y''(2)-2y'(2)\) is equal to

For the differentiable function \(f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\), let \(3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10\), then \(\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|\) is equal to

Let \(f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}\). If \(2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1\) then \(\mathrm{n}\) is equal to ___________

Let f and g be twice differentiable even functions on (\(-\)2, 2) such that \(f\left( {{1 \over 4}} \right) = 0\), \(f\left( {{1 \over 2}} \right) = 0\), \(f(1) = 1\) and \(g\left( {{3 \over 4}} \right) = 0\), \(g(1) = 2\). Then, the minimu...

Let \(x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}\) and \(y(t)=2 \sqrt{2} \sin t \sqrt{\sin 2 t}, t \in\left(0, \frac{\pi}{2}\right)\). Then \(\frac{1+\left(\frac{d y}{d x}\right)^{2}}{\frac{d^{2} y}{d x^{2}}}\) at \(t=\frac{\pi}{4}\) is equal t...

If \({\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| < 2\), then :

If \(y(x) = {\left( {{x^x}} \right)^x},\,x > 0\), then \({{{d^2}x} \over {d{y^2}}} + 20\) at x = 1 is equal to ____________.

For the curve \(C:\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0\), the value of \(3 y^{\prime}-y^{3} y^{\prime \prime}\), at the point \((\alpha, \alpha)\), \(\alpha>0\), on C, is equal to ____________.

Let f : R \(\to\) R satisfy \(f(x + y) = {2^x}f(y) + {4^y}f(x)\), \(\forall\)x, y \(\in\) R. If f(2) = 3, then \(14.\,{{f'(4)} \over {f'(2)}}\) is equal to ____________.

The value of \(\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)\) at \(x=\frac{\pi}{4}\) is

Let f : R \(\to\) R be defined as \(f(x) = {x^3} + x - 5\). If g(x) is a function such that \(f(g(x)) = x,\forall 'x' \in R\), then g'(63) is equal to ________________.

If \(y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} < {x^3} < {{3\pi } \over 2}\), then

If \(y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi } \right)\), then \({{dy} \over {dx}}\) at \(x = {{5\pi } \over 6}\)...

If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then \({{{d^2}y} \over {d{x^2}}}\) at x = 0 is equal to ___________.

Let \(f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)\), 0 < x < 1. Then :

If \(f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)\) and its first derivative with respect to x is \(- {b \over a}{\log _e}2\) when x = 1, where a and b are integers, then the minimum v...

If \(x = 2\sin \theta - \sin 2\theta\) and \(y = 2\cos \theta - \cos 2\theta\),
\(\theta \in \left[ {0,2\pi } \right]\), then \({{{d^2}y} \over {d{x^2}}}\) at \(\theta\) = \(\pi\) is :

Let ƒ and g be differentiable functions on R
such that fog is the identity function. If for some
a, b \(\in\) R, g'(a) = 5 and g(a) = b, then ƒ'(b) is
equal to :

Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1.
If \({{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right)\) and \(y\left( {\sqrt 3 } \right) = {\pi \over 6}\),
then y($${ -...