11. -/2 points details my notes larcalc12 2.3.051. find the derivative of the trigonometric function. y =…

Explanation:

Step1: Recall derivative rules

The derivative of $\csc(x)$ is $-\csc(x)\cot(x)$ and the derivative of $\cos(x)$ is $-\sin(x)$.

Step2: Differentiate term - by - term

For $y = -\csc(x)-\cos(x)$, using the sum - difference rule of derivatives $\frac{d}{dx}(u - v)=\frac{d}{dx}u-\frac{d}{dx}v$. Let $u = -\csc(x)$ and $v=\cos(x)$. Then $\frac{dy}{dx}=\frac{d}{dx}(-\csc(x))-\frac{d}{dx}(\cos(x))$. The derivative of $-\csc(x)$ is $-(-\csc(x)\cot(x))=\csc(x)\cot(x)$ and the derivative of $-\cos(x)$ is $\sin(x)$.

Answer:

$\csc(x)\cot(x)+\sin(x)$