Question

find the derivative of ( f(x) ).

( f(x)=10 cos (x)-4 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, use the constant multiple rule: if $f(x) = c\cdot g(x)$, then $f'(x) = c\cdot g'(x)$.

Step2: Differentiate $10\cos(x)$

For $10\cos(x)$, the derivative is $10\cdot (-\sin(x)) = -10\sin(x)$.

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

For $-4\sin(x)$, the derivative is $-4\cdot \cos(x)$.

Step4: Combine the derivatives

Add the derivatives from Step2 and Step3: $f'(x) = -10\sin(x) - 4\cos(x)$.

Answer:

$-10\sin(x) - 4\cos(x)$