Question

find the derivative of ( f(x) ).
( f(x)=4 cos (x)+sin (x) )
( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules for cosine and sine

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 of the function.

Step2: Differentiate each term of \( f(x) \)

For the first term \( 4\cos(x) \), using the constant multiple rule and the derivative of cosine:
The derivative of \( 4\cos(x) \) is \( 4\times(-\sin(x))=-4\sin(x) \).
For the second term \( \sin(x) \), using the derivative of sine:
The derivative of \( \sin(x) \) is \( \cos(x) \).

Step3: Combine the derivatives

Adding the derivatives of the two terms together, we get \( f'(x)=-4\sin(x)+\cos(x) \).

Answer:

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