Question

find the derivative of ( f(x) ).

( f(x)=-sin (x)-4 cos (x) )

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules for sine and cosine

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Also, the derivative of a constant multiple of a function is the constant multiple of the derivative of the function, and the derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.

Step2: Differentiate $-\sin(x)$

The derivative of $-\sin(x)$ with respect to $x$: using the constant multiple rule (the constant here is -1) and the derivative of $\sin(x)$ is $\cos(x)$, so the derivative of $-\sin(x)$ is $- \cos(x)$.

Step3: Differentiate $-4\cos(x)$

The derivative of $-4\cos(x)$ with respect to $x$: using the constant multiple rule (the constant is -4) and the derivative of $\cos(x)$ is $-\sin(x)$. So we have $-4\times(-\sin(x)) = 4\sin(x)$.

Step4: Combine the derivatives

Since $f(x)=-\sin(x)-4\cos(x)$, by the sum/difference rule of differentiation, $f^\prime(x)$ is the sum of the derivatives of $-\sin(x)$ and $-4\cos(x)$. So $f^\prime(x)=-\cos(x) + 4\sin(x)$.

Answer:

$-\cos(x) + 4\sin(x)$