Mathematics
Differentiation
Previous Years Questions

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^2 f}{d x^2}(x)>0$ for all $x \in(-1,... Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( ... ## MCQ (Single Correct Answer) Let$$f:\left[ {0,2} \right] \to R$$be a function which is continuous on$$\left[ {0,2} \right]$$and is differentiable on$$(0,2)$$with$$f(0)=1$$.... Let$$g(x) = \log f(x)$$, where$$f(x)$$is a twice differentiable positive function on (0,$$\infty$$) such that$$f(x + 1) = xf(x)$$. Then for N = 1... Which of the following is true? Let$$f$$and$$g$$be real valued functions defined on interval$$(-1, 1)$$such that$$g''(x)$$is continuous,$$g\left( 0 \right) \ne 0.g'\lef... If $$f\left( { - 10\sqrt 2 } \right) = 2\sqrt 2 ,$$ then $$f''\left( { - 10\sqrt 2 } \right) =$$ $${{{d^2}x} \over {d{y^2}}}$$ equals Let $$\,\,\,$$$$f\left( x \right) = 2 + \cos x$$ for all real $$X$$. STATEMENT - 1: for eachreal $$t$$, there exists a point $$c$$ in $$\left[ {t,t ... If$$f(x)$$is a twice differentiable function and given that$$f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9$$, then If$$y$$is a function of$$x$$and log$$(x+y)-2xy=0$$, then the value of$$y'(0)$$is equal to Let$$f:\left( {0,\infty } \right) \to R$$and$$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.} $$If$$F\left( {{x^2}} \right) = {x^2}\l... If $${x^2} + {y^2} = 1$$ then If $$y = {\left( {\sin x} \right)^{\tan x}},$$ then $${{dy} \over {dx}}$$ is equal to Let $$f(x)$$ be a quadratic expression which is positive for all the real values of $$x$$. If $$g(x)=f(x)+f''(x)$$, then for any real $$x$$, If $${y^2} = P\left( x \right)$$, a polynomial of degree $$3$$, then $$2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)$$ equals ## Numerical Let $$f\left( \theta \right) = \sin \left( {{{\tan }^{ - 1}}\left( {{{\sin \theta } \over {\sqrt {\cos 2\theta } }}} \right)} \right),$$ where $$- {... ## Subjective If$$\,\,\,y = {{a{x^2}} \over {\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)}} + {{bx} \over {\left( {x - b} \right)\left( {... Find $${{{dy} \over {dx}}}$$ at $$x=-1$$, when $${\left( {\sin y} \right)^{\sin \left( {{\pi \over 2}x} \right)}} + {{\sqrt 3 } \over 2}{\sec ^{ - 1... If$$x = \sec \theta - \cos \theta $$and$$y = {\sec ^n}\theta - {\cos ^n}\theta $$, then show that$$\left( {{x^2} + 4} \right){\left( {{{dy} \ov... If $$\alpha$$ be a repeated root of a quadratic equation $$f(x)=0$$ and $$A(x), B(x)$$ and $$C(x)$$ be polynomials of degree $$3$$, $$4$$ and $$5$$ r... Let $$f$$ be a twice differentiable function such that $$f''\left( x \right) = - f\left( x \right),$$ and $$f'\left( x \right) = g\left( x \right),h... Let$$y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}$$. Find$${{dy} \over {dx}}$$Given$$y = {{5x} \over {3\sqrt {{{\left( {1 - x} \right)}^2}} }} + {\cos ^2}\left( {2x + 1} \right)$$; Find$${{dy} \over {dx}}$$. Find the derivative of$$$f\left( x \right) = \left\{ {\matrix{ {{{x - 1} \over {2{x^2} - 7x + 5}}} & {when\,\,x \ne 1} \cr { - {1 \over ...
Find the derivative of $$\sin \left( {{x^2} + 1} \right)$$ with respect to $$x$$ first principle.

## Fill in the Blanks

If $$x{e^{xy}} = y + {\sin ^2}x,$$ then at $$x = 0,{{dy} \over {dx}} = ..............$$
If $$f\left( x \right) = \left| {x - 2} \right|$$ and $$g\left( x \right) = f\left[ {f\left( x \right)} \right]$$, then $$g'\left( x \right) = .......... The derivative of$${\sec ^{ - 1}}\left( {{1 \over {2{x^2} - 1}}} \right)$$with respect to$$\sqrt {1 - {x^2}} $$at$$x = {1 \over 2}$$is ............ If$${f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3$$are polynomials in$$x$$such that$${f_r}\left( a \right) = {g_r}\...
If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................
If $$y = f\left( {{{2x - 1} \over {{x^2} + 1}}} \right)$$ and $$f'\left( x \right) = \sin {x^2}$$, then $${{dy} \over {dx}} = ..........$$

## True or False

The derivative of an even function is always an odd function.
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