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Continuity and Differentiability

Multiple Choice Questions

431.

$If u = \sin^{- 1} (\frac{x^{4} + y^{4}}{x + y}), then x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = ?$

3u
4u
3sin(u)
3tan(u)

432.

$If y = \frac{\log_{e} (x)}{x} and z = \log_{e} (x), then \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dz} = ?$

$e^{- z}$
$2 e^{- z}$
${ze}^{- z}$
$- e^{- z}$

433.

$If f (x) = \log (e^{x} {(\frac{x - 2}{x + 2})}^{\frac{3}{4}}), then f' (0) = ?$

$\frac{1}{4}$
4
$\frac{- 3}{4}$
1

434.

$If xy \neq 0, x + y \neq 0 and x^{m} y^{n} = {(x + y)}^{m + n}, where m, n not belongs to N, then \frac{dy}{dx} = ?$

$\frac{y}{x}$
$\frac{x + y}{xy}$
xy
$\frac{x}{y}$

435.

$If x^{2} + y^{2} = t + \frac{1}{t}, x^{4} + y^{4} = t^{2} + \frac{1}{t^{2}}, then x^{3} y \frac{dy}{dx} = ?$

436.
If the volume of a sphere increases at the rate of 2 $π$ cm³/ s, then the rate of increase of its radius(in cm/s), when the volume is 288 $π$ cm³, is

$\frac{1}{36}$

$\frac{1}{72}$

$\frac{1}{18}$

$\frac{1}{9}$

$\frac{1}{72}$

$G i v e n, \frac{d V}{d t} = 2 π {cm}^{3} / s ∵ Volume of sphere, V = \frac{4}{3} {πr}^{3} On diffeentiating w . r . t . we get \frac{dV}{dt} = \frac{4}{3} π \times 3 r^{2} \frac{dr}{dt} \Rightarrow 2 π = 4 {πr}^{2} \frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{1}{2 r^{2}} = \frac{1}{2 \times 6^{2}} = \frac{1}{72} cm / s [∵ V = 288 π = \frac{4}{3} {πr}^{3} \Rightarrow 216 = r^{3} \Rightarrow r = 6]$