Question

find the derivative of ( f(x) ).

( f(x)=-2 cos (x)-10 sin (x) )

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules for cos and sin

The derivative of $\cos(x)$ is $-\sin(x)$, and the derivative of $\sin(x)$ is $\cos(x)$. Also, the derivative of a constant multiple of a function is the constant multiple of the derivative.

Step2: Differentiate $-2\cos(x)$

Using the constant multiple rule and the derivative of $\cos(x)$: $\frac{d}{dx}(-2\cos(x))=-2\times(-\sin(x)) = 2\sin(x)$

Step3: Differentiate $-10\sin(x)$

Using the constant multiple rule and the derivative of $\sin(x)$: $\frac{d}{dx}(-10\sin(x))=-10\times\cos(x)= - 10\cos(x)$

Step4: Combine the derivatives

Add the derivatives from Step2 and Step3: $f'(x)=2\sin(x)-10\cos(x)$

Answer:

$2\sin(x) - 10\cos(x)$