Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
-$$y = x ^ { 8 } \sin x$$
A) $8 \sin x \ln x$
B) $8 x ^ { 8 } \sin x \left( \cos x \ln x + \frac { \sin x } { x } \right)$
C) $x \sin x \left( \cos x \ln x + \frac { \sin x } { x } \right)$
D) $8 \cos x \ln x + \frac { \sin x } { x }$

Question

Quizplus · Accepted Answer

Using logarithmic differentiation,

$y = x^8 \sin x$

$\ln y = \ln \left( x^8 \sin x \right)$

$\ln y = 8 \ln x + \ln \left(\sin x\right)$

Differentiating both sides with respect to $x$,

$\frac{1}{y}\frac{dy}{dx} = \frac{8}{x} + \frac{1}{\sin x}\cdot\cos x$

$\frac{dy}{dx} = y\left( \frac{8}{x} + \frac{\cos x}{\sin x} \right)$

Substituting the original expression for $y$,

$\frac{dy}{dx} = x^8\sin x \left( \frac{8}{x} + \frac{\cos x}{\sin x} \right)$

$\frac{dy}{dx} = 8x^8\cos x \ln x + x^7\sin x$

$\frac{dy}{dx} = 8 x ^ { 8 } \sin x \left( \cos x \ln x + \frac { \sin x } { x } \right)$

Therefore, the correct choice is B.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect