Limits and Deri...

lf $$\mathrm{f}(\mathrm{x})$$ is a quadratic expression which is positive for all real vaues of $$\mathrm{x}$$ and $$\mathrm{g}(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{f}'(\mathrm{x})+\mathrm{f}''(\mathrm{x})$$ then for any real value of $$\mathrm{x}$$

Let $$f$$ be a twice differentiable function such that $$f''\left( x \right) =-f\left( x \right) $$ and $$f'(x)=g(x).$$.

If for all $$x, y$$ the function f is defined by; $$f(x)+f(y)+f(x)\cdot f(y)=1$$ and $$f(x) > 0$$.When $$f(x)$$ is differentiable $$f'(x)= $$,

Given, $$f(x)=-\displaystyle \frac {x^3}{3}+x^2 \sin 1.5 a-x \sin a\cdot \sin 2a-5 arc \sin (a^2-8a+17)$$, then

Let f be a twice differentiable such that $$f''(x)=-f(x)$$ and $$f'(x)=g(x)$$. If $$h(x)=\left \{f(x)\right \}^2+\left \{g(x)\right \}^2$$, where $$h(5)=11$$. Find $$h(10)$$

The function $$f(x)=e^x+x$$ being differentiable and one to one, has a differentiable inverse $$f^{-1}(x)$$, then find $$\dfrac {d}{dx} (f^{-1}(x))$$ at the point $$f(log_e 2)$$.

If f (x)$$=\left \{ \displaystyle \frac{|x+2|}{tan^{-1}(_{2}x+2)} \right.x\neq -2$$
$$x=-2,$$ then f(x) is

 Let $$\mathrm{f}(\mathrm{x})=\mathrm{x}+\tan^{-1}\mathrm{x}, \displaystyle \mathrm{g}(\mathrm{x})=\frac{\mathrm{x}}{1+\mathrm{x}^{2}}(\mathrm{x}>0)$$ Then

Find the derivative of $$|x|+a_0x^n+a_1x^{n-1}+a_2x^{n-2}+....+a_{n-1}x+a_n$$

If $$f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}$$, then the value of $$10 f'(102^+)$$ is