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

Let $$f$$ be a twice differentiable function such that

$$f''\left( x \right) = - f\left( x \right),$$ and $$f'\left( x \right) = g\left( x \right),h\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \right]^2}$$

Find $$h\left( {10} \right)$$ if $$h(5)=11$$

A vertical pole stands at a point $$Q$$ on a horizontal ground. $$A$$ and $$B$$ are points on the ground, $$d$$ meters apart. The pole subtends angles $$\alpha $$ and $$\beta $$ at $$A$$ and $$B$$ respectively. $$AB$$ subtends an angle $$\gamma $$ and $$Q$$. Find the height of the pole.

If $$f(x)$$ and $$g(x)$$ are differentiable function for $$0 \le x \le 1$$ such that $$f(0)=2$$, $$g(0)=0$$, $$f(1)=6$$; $$g(1)=2$$, then show that there exist $$c$$ satisfying $$0 < c < 1$$ and $$f'(c)=2g'(c)$$.