If y = $5^{\tan \left(\mathrm{x}\right)}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x} = \frac{\mathrm{\pi }}{4}$ is equal to

• $5\log \left(5\right)$

• $10\log \left(5\right)$

• 0

• ${\left(\log \left(5\right)\right)}^2$

If $\mathrm{y} = {\sin }^{-1}\left(\mathrm{x}\right) \mathrm{and} \mathrm{z} = {\cos }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dz}}$ is equal to

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

• $\frac{1}{2}$

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

• 1

If u = 2(t - sin(t)) and v = 2 (1 - cos(t)), then $\frac{\mathrm{dv}}{\mathrm{du}}$ at t = $\frac{2\mathrm{\pi }}{3}$ is equal to

• $\sqrt{3}$

• $- \sqrt{3}$

• $2\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

14.

If $\mathrm{f}\left(\mathrm{x}\right) = \log \left[{\mathrm{e}}^{\mathrm{x}}{\left(\frac{3 - \mathrm{x}}{3 + \mathrm{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$, then f'(1) is equal to

• $\frac{3}{4}$

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{1}{2}$

If y = ${\left(\log \left(\mathrm{x}\right)\right)}^2$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = e is equal to

• 2

• $\frac{\mathrm{e}}{2}$

• e

• $\frac{2}{\mathrm{e}}$

If y^x = 2^x, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{y}}{\mathrm{x}}\log \left(\frac{2}{\mathrm{y}}\right)$

• $\frac{\mathrm{x}}{\mathrm{y}}\log \left(\frac{2}{\mathrm{y}}\right)$

• $\frac{\mathrm{y}}{\mathrm{x}}\log \left(\frac{\mathrm{y}}{2}\right)$

• $\frac{\mathrm{x}}{\mathrm{y}}\log \left(\frac{\mathrm{y}}{2}\right)$

If x² + 2xy + 2y² = 1, then $\frac{\mathrm{dy}}{\mathrm{dx}}$  at the point where y = 1 is equal to

• 1

• 2

• - 1

• 0

The points on the graph y = x³ - 3x at which the tangent is parallel to x-axis are

• (2, 2) and (1, - 2)

• (- 1, 2) and (- 2, - 2)

• (2, 2) and (- 1, 2)

• (1, - 2) and (- 1, 2)

The slope of the normal to the curve y² - xy - 8 = 0 at the point (0, 2) is equal to

• - 3

• - 6

• 3

• 6

If the straight line y - 2x+ 1 = 0 is the tangent to the curve xy + ax + by = 0 at x = i, then the values of a and b are respectively

• 1 and 2

• 1 and - 1

• - 1 and 2

• 1 and - 2