Question

find the derivative of ( f(x) ).

( f(x)=-9 sin (x)+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, the derivative of a constant multiple of a function is the constant multiple of the derivative.

Step2: Differentiate each term

For the first term $-9\sin(x)$: Using the constant multiple rule and the derivative of $\sin(x)$, we get $-9\times\cos(x)= -9\cos(x)$.
For the second term $\cos(x)$: The derivative of $\cos(x)$ is $-\sin(x)$.

Step3: Combine the derivatives

Adding the derivatives of the two terms together, we have $f'(x)= -9\cos(x)-\sin(x)$.

Answer:

$-9\cos(x) - \sin(x)$