Question

find the derivative of ( f(x) ).

( f(x)=2 sin (x)-4 cos (x) )

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules for sin and cos

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Also, use the constant multiple rule: if $f(x) = c\cdot g(x)$, then $f^\prime(x)=c\cdot g^\prime(x)$.

Step2: Differentiate term by term

For the first term $2\sin(x)$: using the constant multiple rule and derivative of $\sin(x)$, we get $2\cdot\cos(x)$.
For the second term $-4\cos(x)$: using the constant multiple rule and derivative of $\cos(x)$ (which is $-\sin(x)$), we have $-4\cdot(-\sin(x)) = 4\sin(x)$.

Step3: Combine the derivatives

Adding the derivatives of the two terms together: $f^\prime(x)=2\cos(x)+4\sin(x)$.

Answer:

$2\cos(x) + 4\sin(x)$