Find all points of discontinuity of f, where f is defined as following:

$\mathrm{f} ( \mathrm{x} ) = \left\{\begin{array}{l}\begin{array}{c}\left| \mathrm{x} \right| + 3 , \mathrm{x} \le -3 \\ - 2\mathrm{x} , -3 < \mathrm{x} < 3 \end{array}\\ 6\mathrm{x} + 2 , \mathrm{x} \ge 3\end{array}\right.$

$\mathrm{Find} \frac{\mathrm{dy}}{\mathrm{dx}}, \mathrm{if} \mathrm{y} = {\left( \mathrm{cosx}\right)}^{\mathrm{x}} + {\left( \mathrm{sinx} \right)}^{\frac{1}{\mathrm{x}}}$

Find the value of ‘a’ for which the function f defined as

$\mathrm{f} ( \mathrm{x} ) = \left\{\begin{array}{l} \mathrm{a} \sin \frac{\mathrm{\pi }}{2} ( \mathrm{x} + 1 ), \mathrm{x} \le 0\\ \frac{\tan \mathrm{x} - \sin \mathrm{x} }{{\mathrm{x}}^3}, \mathrm{x} > 0 \end{array}\right.$

is continuous at x = 0.

614.

Differentiate  $\mathrm{X}{ }^{\mathrm{x} \cos \mathrm{x}} + \frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 - 1} \mathrm{w}.\mathrm{r}.\mathrm{t}. \mathrm{x}$

If   $\mathrm{x} = \mathrm{a} \left( \mathrm{\theta } - \sin \mathrm{\theta } \right), \mathrm{y} = \left( 1 + \cos \mathrm{\theta } \right), \mathrm{find} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$

If ${\left( \cos \mathrm{x} \right)}^{\mathrm{y}} = {\left( \cos \mathrm{y} \right)}^{\mathrm{x}}, \mathrm{find} \frac{\mathrm{dy}}{\mathrm{dx}}.$

If sin y = x sin (a + y), prove that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\displaystyle {\sin }^2}{\displaystyle }{\displaystyle \mathrm{a}}{\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{y}}}{\sin {\displaystyle }{\displaystyle \mathrm{a}}}$.

If  y = 3 cos ( log x ) + 4 sin ( log x ), show that

${\mathrm{x}}^2 \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

$\mathrm{If} \mathrm{z} = \frac{\mathrm{y}}{\mathrm{x}}\left[\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \cos \left(1 + \frac{\mathrm{y}}{\mathrm{x}} \right) \right], \mathrm{then} \mathrm{x}\frac{\partial \mathrm{z}}{\partial \mathrm{x}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $y\frac{\partial z}{\partial y}$

• $- \mathrm{y}\frac{\partial \mathrm{x}}{\partial \mathrm{y}}$

• $2\mathrm{y}\frac{\partial \mathrm{z}}{\partial \mathrm{y}}$

• $2\mathrm{y}\frac{\partial \mathrm{z}}{\partial \mathrm{x}}$

• 18/e⁴

• 27/e²

• 9/e²

• 9/e²