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Continuity and Differentiability

Long Answer Type

611.

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

$f (x) = \{\begin{cases} \begin{matrix} |x| + 3, x \leq - 3 \\ - 2 x, - 3 < x < 3 \end{matrix} \\ 6 x + 2, x \geq 3 \end{cases}$

612.

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

613.

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

$f (x) = \{\begin{cases} a \sin \frac{π}{2} (x + 1), x \leq 0 \\ \frac{\tan x - \sin x}{x^{3}}, x > 0 \end{cases}$

is continuous at x = 0.

614.

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

615.

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

616.

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

617.
If sin y = x sin (a + y), prove that $\frac{dy}{dx} = \frac{\sin^{2} a + y}{\sin a}$ .

We have,

sin y = x sin ( a + y )

$\Rightarrow x = \frac{\sin y}{\sin (a + y)}$

Differentiating the above function we have,

$1 = \frac{\sin (a + y) \times \cos y \frac{dy}{dx} - \sin y \times \cos (a + y) \frac{dy}{dx}}{\sin^{2} (a + y)} \Rightarrow \sin^{2} (a + y) = [\sin (a + y) \times \cos y - \sin y \times \cos (a + y)] \frac{dy}{dx} \Rightarrow \frac{\sin^{2} (a + y)}{[\sin (a + y) \times \cos y - \sin y \times \cos (a + y)]} = \frac{dy}{dx} \Rightarrow \frac{\sin^{2} (a + y)}{\sin (a + y - y)} = \frac{dy}{dx} \Rightarrow \frac{\sin^{2} (a + y)}{\sin a} = \frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{\sin^{2} (a + y)}{\sin a}$