Let $$f:R \to R,\,g:R \to R$$ and $$h:R \to R$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \rig...

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 $$g\left( x \right) = \log f\left( x \right)$$ where $$f(x)$$ is twice differentible positive function on $$\left( {0,\infty } \right)$$ such that...

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...

$${{{d^2}x} \over {d{y^2}}}$$ 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

Let $$f\left( \theta \right) = \sin \left( {{{\tan }^{ - 1}}\left( {{{\sin \theta } \over {\sqrt {\cos 2\theta } }}} \right)} \right),$$ where $$ - {...

If the function $$f\left( x \right) = {x^3} + {e^{{x \over 2}}}$$ and $$g\left( x \right) = {f^{ - 1}}\left( x \right)$$, then the value of $$g'(1)$$ ...

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.

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 $${{dy} \over {dx}} = ..........$$

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