Differentiation

Let $\mathbb{R}$ denote the set of all real numbers. Consider the polynomial function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$ f(x)=\frac{d^{10}}{d x^{10}}\left(\left(x^2-1\right)^{10}\right), \quad \text { for all } x \in \math...

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 $x \in(-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in(-1,1)$ for which $f(x)=x$. Th...

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 \(f\left( x \right)= {x^3} + 3x + 2,\) \(g\left( {f\left( x \right)} \right) = x\) and $$h\left( {g\...

Let \(F:R \to R\) be a thrice differentiable function. Suppose that
\(F\left( 1 \right) = 0,F\left( 3 \right) = - 4\) and \(F'\left( x \right) < 0\) for all \(x \in \left( {{1 \over 2},3} \right).\) Let $$f\left( x \right) = xF\left( x \r...

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
\(F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)dt}\) for $...

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

Which of the following is true?

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, 2, 3, ..., \(g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) =\)

Let \(f\) and \(g\) be real valued functions defined on interval \((-1, 1)\) such that \(g''(x)\) is continuous, \(g\left( 0 \right) \ne 0.\) \(g'\left( 0 \right) = 0\), \(g''\left( 0 \right) \ne 0\), and $$f\left( x \right) = g\left( x \ri...

If \(f\left( { - 10\sqrt 2 } \right) = 2\sqrt 2 ,\) then \(f''\left( { - 10\sqrt 2 } \right) =\)

\(\frac{d^{2} x}{d y^{2}}\) equals :

Let \(\,\,\,\)\(f\left( x \right) = 2 + \cos x\) for all real \(X\).
STATEMENT - 1: for eachreal \(t\), there exists a point \(c\) in \(\left[ {t,t + \pi } \right]\) such that \(f'\left( c \right) = 0\) because
STATEMENT - 2: $$f\left( t ...

\({{{d^2}x} \over {d{y^2}}}\) equals

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 \(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable function such that \(|f(x)| \leq 1\) and \(f'(x)=g(x)\), where,\(f^{2}(0)+g^{2}(0)=9\) then prove that there exists some \(c \in(-3,3)\) such that $$g(c) \circ...

\(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable
function such that \(\left| {f\left( x \right)} \right| \le 1\) and \(f'(x)=g(x).\)
If \({f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.\) Prove that there e...

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 \(F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)\), then \(f(4)\) equals

If \({x^2} + {y^2} = 1\) then

If\(\,\,\,\) \(y = {{a{x^2}} \over {\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)}} + {{bx} \over {\left( {x - b} \right)\left( {x - c} \right)}} + {c \over {x - c}} + 1\),
prove that $${{y'} \over y} = {1 \over x}\lef...

If \(x{e^{xy}} = y + {\sin ^2}x,\) then at \(x = 0,{{dy} \over {dx}} = ..............\)

If \(y = {\left( {\sin x} \right)^{\tan x}},\) then \({{dy} \over {dx}}\) is equal to

Find \({{{dy} \over {dx}}}\) at \(x=-1\), when
\({\left( {\sin y} \right)^{\sin \left( {{\pi \over 2}x} \right)}} + {{\sqrt 3 } \over 2}{\sec ^{ - 1}}\left( {2x} \right) + {2^x}\tan \left( {In\left( {x + 2} \right)} \right) = 0\)

If \(f\left( x \right) = \left| {x - 2} \right|\) and \(g\left( x \right) = f\left[ {f\left( x \right)} \right]\), then \(g'\left( x \right) = ...............\) for \(x > 20\)

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 \(x = \sec \theta - \cos \theta\) and \(y = {\sec ^n}\theta - {\cos ^n}\theta\), then show
that \(\left( {{x^2} + 4} \right){\left( {{{dy} \over {dx}}} \right)^2} = {n^2}\left( {{y^2} + 4} \right)\)

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

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 \({f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3\)
and $$F\left( x \right) = \left| {\ma...

If \(f\left( x \right) = {\log _x}\left( {In\,x} \right),\) then \(f'\left( x \right)\) at \(x=e\) is ................

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 \(5\) respectively,
then show that $$\left| {\matrix{
{A\left( x \right)} & {B\left( x \right...

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

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\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \r...

If \(y = f\left( {{{2x - 1} \over {{x^2} + 1}}} \right)\) and \(f'\left( x \right) = \sin {x^2}\), then \({{dy} \over {dx}} = ..........\)

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
\($f\left( x \right) = \left\{ {\matrix{ {{{x - 1} \over {2{x^2} - 7x + 5}}} & {when\,\,x \ne 1} \cr { - {1 \over 3}} & {when\,\,x = 1} \cr } } \right.\)$
at \(x=1\)

Find the derivative of \(\sin \left( {{x^2} + 1} \right)\) with respect to \(x\) first principle.