Find $\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)$ .
A) $\frac { 1 } { \sin ^ { 2 } \theta }$
B) $- \frac { 1 } { \sin ^ { 2 } \theta }$
C) $- \frac { \sin \theta } { \cos \theta }$
D) $1 - \frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta }$

Question

Quizplus · Accepted Answer

The derivative of $\frac{\cos\theta}{\sin\theta}$ can be found using the quotient rule for derivatives, $ \frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'g - fg'}{g^2} $, where $f = \cos\theta$ and $g = \sin\theta$. Thus, $f' = -\sin\theta$ and $g' = \cos\theta$. Substituting these into the quotient rule gives $\frac{-\sin\theta \cdot \sin\theta - \cos\theta \cdot \cos\theta}{\sin^2\theta} = -\frac{\sin^2\theta + \cos^2\theta}{\sin^2\theta} = -\frac{1}{\sin^2\theta}$, using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$.

Find ddθ(cos⁡θsin⁡θ)\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)dθd​(sinθcosθ​)

Find $\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)$