JEE MAIN · MATHEMATICS · DIFFERENTIATION
JEE Main Differentiation Questions & Solutions
55 solved questions on Differentiation, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
55 solved questions on Differentiation, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
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 :
View solution →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…
View solution →Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If $${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)}…
View solution →If y = $$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}} $$, then ${{dy} \over {dx}}$ at x = 0 is _______.
View solution →Let $$f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$$. Then $f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7…
View solution →Suppose for a differentiable function $h, h(0)=0, h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $g(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$, then $g^{\prime}(0)$ is…
View solution →If $$\left( {a + \sqrt 2 b\cos x} \right)\left( {a - \sqrt 2 b\cos y} \right) = {a^2} - {b^2}$$ where a > b > 0, then ${{dx} \over {dy}}\,\,at\left( {{\pi \over 4},{\pi \over 4}}…
View solution →Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.
View solution →Let $$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)$$. Then, at x = 1,
View solution →$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1
View solution →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…
View solution →If $$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi }…
View solution →If $y(x) = {\left( {{x^x}} \right)^x},\,x 0$, then ${{{d^2}x} \over {d{y^2}}} + 20$ at x = 1 is equal to ____________.
View solution →Let $$f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$$, 0 < x < 1. Then :
View solution →Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of…
View solution →If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then ${{{d^2}y} \over {d{x^2}}}$ at x = 0 is equal to ___________.
View solution →If $$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$ and its first derivative with respect to x is $- {b \over a}{\log _e}2$…
View solution →Let $x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}$ and $y(t)=2 \sqrt{2} \sin t \sqrt{\sin 2 t}, t \in\left(0, \frac{\pi}{2}\right)$. Then $\frac{1+\left(\frac{d y}{d…
View solution →Let $f$ and $g$ be the twice differentiable functions on $\mathbb{R}$ such that $f''(x)=g''(x)+6x$ $f'(1)=4g'(1)-3=9$ $f(2)=3g(2)=12$. Then which of the following is NOT true?
View solution →Let $f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}$. If $2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1$ then $\mathrm{n}$ is equal to ___________
View solution →If $$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…
View solution →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 :
View solution →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
View solution →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…
View solution →If $$y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi }…
View solution →Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _\limits{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$.…
View solution →Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$. If $f'(0)=2$, then $|f(-2)|$ is equal to…
View solution →If $f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)$, then the value of $f(4)-g(4)$ is equal to ____________.
View solution →Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8…
View solution →Let $g: \mathbf{R} \rightarrow \mathbf{R}$ be a non constant twice differentiable function such that…
View solution →If $f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$, then
View solution →For the curve $C:\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0$, the value of $3 y^{\prime}-y^{3} y^{\prime \prime}$, at the point $(\alpha, \alpha)$, $\alpha0$, on…
View solution →If $x = 2\sin \theta - \sin 2\theta$ and $y = 2\cos \theta - \cos 2\theta$, $\theta \in \left[ {0,2\pi } \right]$, then ${{{d^2}y} \over {d{x^2}}}$ at $\theta$ = $\pi$ is :
View solution →Let f and g be twice differentiable even functions on ($-$2, 2) such that $f\left( {{1 \over 4}} \right) = 0$, $f\left( {{1 \over 2}} \right) = 0$, $f(1) = 1$ and $g\left( {{3…
View solution →If $y(x)=x^{x},x 0$, then $y''(2)-2y'(2)$ is equal to
View solution →$$\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 }$$
View solution →Let $f(x) = 2x + {\tan ^{ - 1}}x$ and $g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$. Then
View solution →Let $f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}$ For $\mathrm{n} \geq 2$, define $f^{\mathrm{n}}(x)=f^{1} \mathrm{o} f^{\mathrm{n}-1}(x)$. If…
View solution →The value of $\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$ at $x=\frac{\pi}{4}$ is
View solution →Let y = y(x) be a function of x satisfying $y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}}$ where k is a constant and $y\left( {{1 \over 2}} \right) = - {1 \over 4}$. Then ${{dy}…
View solution →Let xk + yk = ak, (a, k > 0 ) and ${{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0$, then k is:
View solution →If $$y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2}
View solution →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
View solution →If $2 x^{y}+3 y^{x}=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to :
View solution →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
View solution →If y2 + loge (cos2x) = y, $x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$, then :
View solution →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:
View solution →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 ____________.
View solution →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
View solution →The derivative of ${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with respect to ${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$…
View solution →If $${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y|
View solution →Let $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then $f^{\prime}(10)$ is equal to ____________.
View solution →Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(x) \geq 0, f^{\prime\prime}(x)=f(x), f(0)=0, f^{\prime}(0)=3$. Then $9…
View solution →Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all…
View solution →$$ \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…
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