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

Question

Quizplus · Accepted Answer

To differentiate $y = ( \sin x ) ^ { \cos x }$ using logarithmic differentiation, first take the natural logarithm of both sides: $\ln(y) = \ln((\sin x)^{\cos x})$. This simplifies to $\ln(y) = \cos x \ln(\sin x)$. Differentiating both sides with respect to $x$ gives $\frac{1}{y} \frac{dy}{dx} = -\sin x \ln(\sin x) + \cos x \cot x$. Multiplying both sides by $y = (\sin x)^{\cos x}$ gives the derivative of $y$ with respect to $x$ as $(\sin x)^{\cos x}(\cos x \cot x - \sin x \ln(\sin x))$.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect