Question

find the derivative of ( f(x) ).

( f(x)=sin (x)+cos (x) )

( f^{prime}(x)= )

Explanation:

Step1: Recall derivative rules

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Also, use the sum rule for derivatives: if $f(x)=u(x)+v(x)$, then $f^\prime(x)=u^\prime(x)+v^\prime(x)$.

Step2: Apply rules to $f(x)$

For $f(x)=\sin(x)+\cos(x)$, let $u(x)=\sin(x)$ and $v(x)=\cos(x)$. Then $u^\prime(x)=\cos(x)$ and $v^\prime(x)=-\sin(x)$. So $f^\prime(x)=\cos(x)+(-\sin(x))=\cos(x)-\sin(x)$.

Answer:

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