Find the derivative of y with respect to the independent variable.
-$$y = 3^{ \cos \pi \theta}$$
A) $\pi 3 ^ { \cos \pi \theta } \ln 3$
B) $3 ^{\cos \pi \theta}$
C) $- 3 ^{\cos \pi \theta} \ln 3 \sin \pi \theta$
D) $- \pi 3 ^ { \cos \pi \theta } \ln 3 \sin \pi \theta$

Question

Quizplus · Accepted Answer

To find the derivative of $y = 3^{\cos \pi \theta}$ with respect to $\theta$, we use the chain rule. The derivative of $3^u$ with respect to $u$ is $3^u \ln 3$, and the derivative of $\cos \pi \theta$ with respect to $\theta$ is $-\pi \sin \pi \theta$. Applying the chain rule, we get $\frac{dy}{d\theta} = 3^{\cos \pi \theta} \ln 3 \cdot (-\pi \sin \pi \theta) = -\pi 3^{\cos \pi \theta} \ln 3 \sin \pi \theta$.

Find the Derivative of Y with Respect to the Independent y=3cos⁡πθy = 3^{ \cos \pi \theta}y=3cosπθ

Find the Derivative of Y with Respect to the Independent $y = 3^{ \cos \pi \theta}$