Differentiation...

If $$ \displaystyle {f}'\left ( x \right )=g\left ( x \right ) $$ and $$ \displaystyle {g}'\left ( x \right )=-f\left ( x \right ) $$ and $$ \displaystyle f\left ( 2 \right )=4={f}'\left ( 2 \right ) $$ then $$ \displaystyle f^{2}\left ( 16 \right )+g^{2}\left ( 16 \right ) $$ is

If $$ \displaystyle y=\sec ^{ -1 }{ \left( \frac { x+1 }{ x-1 }  \right)  } +\sin ^{ -1 }{ \left( \frac { x-1 }{ x+1 }  \right)  }  $$ then $$ \displaystyle \frac{dy}{dx} $$ is equal to

Let $$f\left( x \right)=\sqrt { x-1 } +\sqrt { x+24-10\sqrt { x-1 }  } ;1<x<26$$ be a real valued function. Then $$f'(x)$$ for $$1<x<26$$ is

Find the area of the triangle whose vertices are $$(3,2), \ (-2, -3)$$ and $$(2,3)$$.

If for a non-zero $$x,$$ the function $$f(x)$$ satisfies the equation $$\displaystyle af\left( x \right)+bf\left( \frac { 1 }{ x }  \right) =\frac { 1 }{ x } -5\left( a\neq b \right) $$ then $$f'(x)$$ is equal to

If $${ S }_{ n }$$ denotes the sum of $$n$$ terms of $$g.p$$. whose common ratio is $$r$$, then $$\displaystyle \left( r-1 \right) \frac { d{ S }_{ n } }{ dr } $$ is equal to

Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are $$(2,2)$$, $$(4,4)$$ and $$(2,6)$$.

Find $$ \displaystyle  \frac{dy}{dx}$$ if $$ y=x^{x} $$

$$ \displaystyle f^{ ' }\left( x \right) =g\left( x \right) $$ and $$ \displaystyle g^{ ' }\left( x \right) =-f\left( x \right)$$ for all real x and $$ \displaystyle f\left( 5 \right) =2=f^{ ' }\left( 5 \right) $$ then $$ \displaystyle f^{ 2 }\left( 10 \right) +g^{ 2 }\left( 10 \right) $$ is -

If $$y = \displaystyle \tan^{-1}\left (\cot x \right ) +\cot^{-1}(\tan x),$$ then $$\displaystyle \frac{dy}{dx}$$ is equal to-