Continuity and ...

 $$\displaystyle\frac{dy}{dx}$$ at $$\displaystyle t=\frac{\pi}{4}$$ for $$\displaystyle x=a\left[\cos{t}+\frac{1}{2}\log{\tan^2{\frac{t}{2}}}\right]$$ and $$y=a\sin{t}$$ is

The relation between the parameter '$$t$$' and the angle $$\alpha$$ between the tangent to the given curve and the $$x-$$axis is given by, '$$t$$' equals to

If $$y={(\tan{x})}^{\displaystyle{(\tan{x})}^{\displaystyle\tan{x}}}$$, then find $$\displaystyle\frac{dy}{dx}$$ at $$\displaystyle x=\frac{\pi}{4}$$.

If $$x=a\sec^{3}{\theta}$$ and $$y=a\tan^{3}{\theta}$$, then $$\displaystyle\frac{dy}{dx}$$ at $$\theta=\displaystyle\frac{\pi}{3}$$ is 

Find $$\displaystyle\frac{dy}{dx}$$ if $$x=a(\theta-\sin{\theta})$$ and $$y=a(1-\cos{\theta})$$.

If   $$\displaystyle y^{x}=x^{\sin y} $$, find $$\cfrac{dy}{dx}$$.

Discuss the applicability of Rolle's theorem to $$\displaystyle f(x)=\log \left[\frac{x^{2}+ab}{(a+b)x}\right],$$ in the interval$$ [a,b].$$

Differentiate $$\displaystyle x^{\sin^{-1}x}$$ w.r.t. $$\displaystyle \sin ^{-1}x.$$

$$\displaystyle y=(\cot x)^{\sin x}+(\tan  x)^{\cos x}$$.Find dy/dx 

Verify Rolle's theorem for $$\displaystyle f(x)=x(x+3)e^{-x/2}$$ in $$(-3,0)$$