Question

find the derivative of ( f(x) ).

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

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules

The derivative of \(\cos(x)\) is \(-\sin(x)\), and the derivative of \(\sin(x)\) is \(\cos(x)\). Also, the derivative of a difference of functions is the difference of their derivatives.

Step2: Differentiate term by term

For \(f(x)=\cos(x)-\sin(x)\), we differentiate \(\cos(x)\) and \(-\sin(x)\) separately.
The derivative of \(\cos(x)\) is \(-\sin(x)\), and the derivative of \(-\sin(x)\) is \(-\cos(x)\) (since the derivative of \(\sin(x)\) is \(\cos(x)\) and we have a negative sign).
So, \(f^{\prime}(x)=\frac{d}{dx}[\cos(x)]-\frac{d}{dx}[\sin(x)] = -\sin(x)-\cos(x)\)

Answer:

\(-\sin(x) - \cos(x)\)