Differentiation...

 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

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 $$f'(x)=g(x).$$.

Let $$f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 \le x \le 26$$ be a real valued function, then $$f'(x)$$ for $$1 < x < 26$$ is

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 $$h(10)$$

Let $$f$$ be a differentiable function satisfying $$f(x) + f(y) + f(z) + f(x)f(y)f(z) = 14$$ for all $$x,\space y,\space z \in R$$
Then,

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 $$f(log_e 2)$$.

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

Suppose, $$A=\displaystyle \frac {dy}{dx}$$ of $$x^2+y^2=4$$ at $$(\sqrt 2, \sqrt 2), B=\displaystyle \frac {dy}{dx}$$ of $$sin y+sin x=sin x\cdot sin y$$ at $$(\pi, \pi)$$ and $$C=\displaystyle \frac {dy}{dx}$$ of $$2e^{xy}+e^xe^y-e^x=e^{xy+1}$$ at $$(1, 1)$$, then $$(A-B-C)$$ has the value equal to .....