Differentiation

If \(y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi } \over 4},\pi } \right)\)
$${{dy} \over {d\alpha }}\,\,at\...

Let xk + yk = ak, (a, k > 0 ) and \({{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0\), then k is:

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 \({{dy} \over {dx}}\) at x = \({1 \over 2}\), is equal to :

The derivative of
\({\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)\) with respect to \({\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)\) at x = \({1 \over 2}\) is :

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 \({{dx} \over {dy}}\,\,at\left( {{\pi \over 4},{\pi \over 4}} \right)\) is :

If y2 + loge (cos2x) = y, \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\), then :

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 _______.

If   x \(=\) 3 tan t and y \(=\) 3 sec t, then the value of \({{{d^2}y} \over {d{x^2}}}\) at t \(= {\pi \over 4},\) is :

If \(2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}\),
x \(\in\) \(\left( {0,{\pi \over 2}} \right)\) then \(dy \over dx\) is equal to:

If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of
ƒ(ƒ(ƒ(x))) + (ƒ(x))2
at x = 1 is :

For x > 1, if (2x)2y = 4e2x\(-\)2y,
then (1 + loge 2x)2 \({{dy} \over {dx}}\) is equal to :

If ey
+ xy = e, the ordered pair \(\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)\) at x = 0 is equal to :

The derivative of \({\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)\), with respect to \({x \over 2}\)
, where \(\left( {x \in \left( {0,{\pi \over 2}} \right)} \right)\) is :

If  xloge(logex) \(-\) x2 + y2 = 4(y > 0), then \({{dy} \over {dx}}\) at x = e is equal to :

Let f : R \(\to\) R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x \(\in\) R. Then f(2) equals -

Let f(x) = loge(sin x), (0 < x < \(\pi\)) and g(x) = sin–1
(e–x
), (x \(\ge\) 0). If \(\alpha\) is a positive real number such that
a = (fog)'(\(\alpha\)) and b = (fog)(\(\alpha\)), then :

If \(x = \sqrt {{2^{\cos e{c^{ - 1}}}}}\) and \(y = \sqrt {{2^{se{c^{ - 1}}t}}} \,\,\left( {\left| t \right| \ge 1} \right),\) then \({{dy} \over {dx}}\) is equal to :

If \(f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \cr } } \right|,\) then \(\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}\)

If   x2 + y2 + sin y = 4, then the value of \({{{d^2}y} \over {d{x^2}}}\) at the point (\(-\)2,0) is :

If    f(x) = sin-1 \(\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right),\) then f'\(\left( { - {1 \over 2}} \right)\) equals :

Let f be a polynomial function such that
f (3x) = f ' (x) . f '' (x), for all x \(\in\) R. Then :

If y = \({\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},\)
then (x2 \(-\) 1) \({{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}\) is equal to :

If for \(x \in \left( {0,{1 \over 4}} \right)\), the derivatives of
\({\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right)\) is \(\sqrt x .g\left( x \right)\), then \(g\left( x \right)\) equals

If \(g\) is the inverse of a function \(f\) and \(f'\left( x \right) = {1 \over {1 + {x^5}}},\) then \(g'\left( x \right)\) is equal to:

If \(y = \sec \left( {{{\tan }^{ - 1}}x} \right),\) then \({{{dy} \over {dx}}}\) at \(x=1\) is equal to :

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

Let \(f:\left( { - 1,1} \right) \to R\) be a differentiable function with \(f\left( 0 \right) = - 1\) and \(f'\left( 0 \right) = 1\). Let \(g\left( x \right) = {\left[ {f\left( {2f\left( x \right) + 2} \right)} \right]^2}\). Then $$g'\lef...

Let \(y\) be an implicit function of \(x\) defined by \({x^{2x}} - 2{x^x}\cot \,y - 1 = 0\). Then \(y'(1)\) equals

If \({x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},\) then \({{{dy} \over {dx}}}\) is

If \(x = {e^{y + {e^y} + {e^{y + .....\infty }}}}\) , \(x > 0,\) then \({{{dy} \over {dx}}}\) is

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

Let \(f\left( x \right)\) be a polynomial function of second degree. If \(f\left( 1 \right) = f\left( { - 1} \right)\) and \(a,b,c\) are in \(A.P,\) then \(f'\left( a \right),f'\left( b \right),f'\left( c \right)\) are in

If \(y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},\) then \(\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}\) is