Continuity and ...

If $$xlog_e(log_ex)-x^2+y^2=4(y>0)$$, then dy/dx at $$x=e$$ is equal to:

If $$x=\cos^3\theta $$ and $$y=\sin^3\theta$$, then $$1+\left(\displaystyle\frac{dy}{dx}\right)^2$$ is equal to:

For the curve $$x = t^2 - 1, y = t^2 - t$$, tangent is parallel to $$x$$ - axis where,

$$\displaystyle \frac{d}{dx}(\tan ^{-1}x)$$

$$\displaystyle \frac{d(sin^{-1}x)}{dx}$$

If the tangent to the curve $$x=a \, (\theta + \sin \, \theta), y=a (1+ \cos \,\theta)$$ at $$ \theta=\dfrac{\pi}{3}$$ makes an angle $$\alpha (0 \leq\alpha < \pi)$$ with x-axis, then $$\alpha$$ =

If $$y$$ is expressed in terms of a variable $$x$$ as $$y = f(x)$$, then $$y$$ is called

If $$f(x)=\begin{cases}
ax &a<1&\\
ax^2+bx+2 &a\ge 1&.
\end{cases}$$
Then the values of $$a$$, $$b$$ for which $$f(x)$$ is differentiable, are

Let $$f(x)=x^2e^x$$ then $$f''(0)=$$

If $$f(x)=\dfrac{1}{2}\left [ \left | \sin x \right |+\sin x \right ],\ 0 < x \leq 2\pi$$, then $$f$$ is