find the derivative of (cos\theta-\frac{sin\theta}{4})

Explanation:

Step1: Recall derivative rules

Use the sum - difference rule of derivatives $(\ f(x)\pm g(x))'=f'(x)\pm g'(x)$ and the basic derivative formulas $(\cos x)' =-\sin x$, $(\sin x)'=\cos x$.

Step2: Differentiate each term

The derivative of $\cos\theta$ is $-\sin\theta$, and the derivative of $-\frac{\sin\theta}{4}$ is $-\frac{1}{4}(\sin\theta)'$. Since $(\sin\theta)'=\cos\theta$, the derivative of $-\frac{\sin\theta}{4}$ is $-\frac{1}{4}\cos\theta$.

Step3: Combine the results

The derivative of $\cos\theta-\frac{\sin\theta}{4}$ is $-\sin\theta-\frac{1}{4}\cos\theta$.

Answer:

$-\sin\theta-\frac{1}{4}\cos\theta$