Continuity and ...

If $$y=(x^{x})^{x}$$ then $$\dfrac {dy}{dx}=$$

If $$ x=a(\cos\theta+log\ \tan\dfrac{\theta}{2})$$ and $$y=a\sin\theta$$, then$$\dfrac{dy}{dx}$$ is equal to

If $$f(x)=\dfrac{a^x}{x^a}$$ then $$f'(a)=$$?

If $$y=\tan^{-1}x+\cot^{-1}x+\sec^{-1}x+\csc^{-1}x$$,then $$\dfrac {dy}{dx}$$ is equal to

Solve this:-$$\dfrac{d}{{dx}}\left( {\tan^{ - 1}\left( {\sinh \,X} \right)} \right) = $$

If $$f(x)=\sin^4x+\cos^4x$$. Then $$f$$ is an increasing function in the interval

Let $$f(x)$$ be a function defined on$$(-a,a)$$ with $$a > 0$$. Amuse that $$f(x)$$ is continuous at $$x=0$$ and $$\underset{x\to 0}{\lim}\dfrac {f(x)-f(kx)}{x}=\alpha$$, where $$k \in (0,1)$$ then compute $$f'(0^{+})$$ and $$f'(0^{-})$$, and comment upon the differentiablity of $$f$$ at $$x=0$$? Denote $$\alpha$$.

Solve:

If $$x=a\left(\cos t+\log \tan\dfrac{t}{2}\right), y=a\sin t$$, then evaluate $$\dfrac{d^2y}{dx^2}$$ at $$t=\dfrac{\pi}{3}$$.

The derivative of $$\sin^{-1}\left (\dfrac {2x}{1 + x^{2}}\right )$$ with respect to $$\tan^{-1} \left (\dfrac {2x}{1 - x^{2}}\right )$$is