Question

1. if ( f(x)=x+sin x ), then ( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(x\)

The derivative of \(x\) with respect to \(x\) is \(1\).

Step2: Differentiate \(\sin x\)

The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\).

Step3: Apply sum rule of differentiation

If \(f(x)=u(x)+v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)\). Here \(u(x) = x\) and \(v(x)=\sin x\). So \(f^{\prime}(x)=\frac{d}{dx}(x)+\frac{d}{dx}(\sin x)\).

Answer:

\(1 + \cos x\)