If y = ${\tan }^{-1}\left(\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 1/2

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• 1

The derivative of ${\cos }^{-1}\left(2{\mathrm{x}}^2 - 1\right) \mathrm{w}.\mathrm{r}.\mathrm{t}. {\cos }^{-1}\left(\mathrm{x}\right)$ is

• 1 - x²

• $\frac{2}{\mathrm{x}}$

• $\frac{1}{2\sqrt{1 - {\mathrm{x}}^2}}$

• 2

The value of c in mean value theorem for the function f(x) = x² in [2, 4] is

• 3

• 7/2

• 4

• 2

If the function f(x) = $$\begin{gathered} \left\{1 + \sin \left(\frac{\mathrm{\pi x}}{2}\right), \mathrm{for} - \infty < \mathrm{x} \le 1 \\ \mathrm{ax} +\hspace{0.17em}\mathrm{b}, \mathrm{for} 1 < \mathrm{x} <\hspace{0.17em}3 \\ 6\tan \left(\frac{{\displaystyle \mathrm{\pi x}}}{{\displaystyle 12}}\right), \mathrm{for} 3 \le \mathrm{x} < 6\right\} \end{gathered}$$ is continuous in the interval (- $\infty$, 6), then the values of a and b are respectively

• 0, 2

• 1, 1

• 2, 0

• 2, 1

945.

The value of m for which the function f(x) = $$\begin{gathered} \left\{{\mathrm{mx}}^2, \mathrm{x} \le 1 \\ 2\mathrm{x}, \mathrm{x} > 1\right\} \end{gathered}$$, is differentiable at x = 1, is

• 0

• 1

• 2

• does not exist

If y = (1 + x^1/4)(1 + x^1/2)(1 - x^1/4), then dy/dx is equal to

• 1

• - 1

• x

• $\sqrt{\mathrm{x}}$

If y = $\log \left(\log \left(\mathrm{x}\right)\right), \mathrm{then} {\mathrm{e}}^{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{\mathrm{xlog}\left(\mathrm{x}\right)}$

• $\frac{1}{\mathrm{x}}$

• $\frac{1}{\log \left(\mathrm{x}\right)}$

• e^y

For the function f(x) = x² - 6x + 8, $2 \le \mathrm{x} \le 4$, the value of x for which f'(x) vanishes, is

• 9/4

• 5/2

• 3

• 7/2

If $\mathrm{y} = {\left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)}^{\mathrm{n}}$, then $\left(1 + {\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• n²y

• - n²y

• - y

• 2x²y

If x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{1 + \mathrm{x}}{1 + \log \left(\mathrm{x}\right)}$

• $\frac{1 - \log \left(\mathrm{x}\right)}{1 + \log \left(\mathrm{x}\right)}$

• not defined

• $\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$