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Limits And Derivatives

Multiple Choice Questions

$\lim_{x \to 0} \frac{1 - \cos^{3} (4 x)}{x^{2}} = ?$

$If f (x) = \sin (\cos (x)), then f' (x) = ?$

$\cos (\cos (x))$
$\sin (- \sin (x))$
$(\sin (x)) \cos (\cos (x))$
$(- \sin (x)) \cos (\cos (x))$

$What is the derivative of the function f (x) = e^{\tan (x)} + \ln (\sec (x)) - e^{\ln (x)} at x = \frac{π}{4} ?$

$\frac{e}{2}$
e
2e
4e

$What is \lim_{x \to 0} \frac{\tan (x)}{\sin (2 x)} = ?$

$\frac{1}{2}$
1
2
Limit does not exist

$What is \lim_{h \to 0} \frac{\sqrt{2 x + 3 h} - \sqrt{2 x}}{2 h} = ?$

$\frac{1}{2 \sqrt{2} x}$
$\frac{3}{\sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$

What is the value of $\log_{7} (\sqrt{7 \sqrt{7 \sqrt{7}}}) = ?$

$3 \log_{2} (7)$
$1 - 3 \log_{2} (7)$
$1 - 3 \log_{7} (2)$
$\frac{7}{8}$

$If f (x) = \sqrt{25 - x^{2}}, then what is \lim_{x \to 1} \frac{f (x) - f (1)}{x - 1} = ?$

$- \frac{1}{\sqrt{24}}$
$\frac{1}{\sqrt{24}}$
$- \frac{1}{4 \sqrt{3}}$
$\frac{1}{4 \sqrt{3}}$

$What is \lim_{x \to 0} \frac{\sqrt{1 - \cos (θ)}}{θ} = ?$

$\sqrt{2}$
$2 \sqrt{2}$
$\frac{1}{\sqrt{2}}$
$- \frac{1}{2 \sqrt{2}}$

9.
$If y = {|\sin (x)|}^{|x|}, then what is the value of \frac{dy}{dx} at x = - \frac{π}{6} ?$

$\frac{2^{- \frac{π}{6}} (6 \ln (2) - \sqrt{3} π)}{6}$

$\frac{2^{\frac{π}{6}} (6 \ln (2) + \sqrt{3} π)}{6}$

$\frac{2^{- \frac{π}{6}} (6 \ln (2) + \sqrt{3} π)}{6}$

$\frac{2^{\frac{π}{6}} (6 \ln (2) - \sqrt{3} π)}{6}$

$\frac{2^{- \frac{π}{6}} (6 \ln (2) - \sqrt{3} π)}{6}$

$y = {|\sin (x)|}^{|x|} y = {(- \sin (x))}^{- x} \log (y) = (- x) \log (- \sin (x)) \frac{1}{y} . \frac{dy}{dx} = \frac{- x}{- \sin (x)} (- \cos (x)) + \log (- \sin (x)) (- 1) \frac{dy}{dx} = {(- \sin (x))}^{- x} [\frac{x}{\sin (x)} (- \cos (x)) - \log (- \sin (x))] {(\frac{dy}{dx})}_{x = - \frac{π}{6}} = {(\frac{1}{2})}^{^{\frac{π}{6}}} [- \frac{π}{6} \sqrt{3} - \log (\frac{1}{2})] = 2^{- \frac{π}{6}} (\frac{6 \log (2) - \sqrt{3} π}{6})$