Question

antiderivatives: problem 3
(1 point)
find the derivative of ( f(x)=x sin (x)+cos (x)+c ) to complete the following integration formula:
( int square d x=x sin (x)+cos (x)+c )
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Explanation:

Step1: Apply the sum rule of differentiation

The sum rule states that \((u + v+w)^\prime=u^\prime + v^\prime+w^\prime\). Let \(u = x\sin(x)\), \(v=\cos(x)\), and \(w = C\). Then \(f^\prime(x)=(x\sin(x))^\prime+(\cos(x))^\prime+(C)^\prime\).

Step2: Apply the product rule to \((x\sin(x))^\prime\)

The product rule is \((uv)^\prime = u^\prime v+uv^\prime\). Here \(u = x\), \(u^\prime=1\), \(v=\sin(x)\), \(v^\prime=\cos(x)\). So \((x\sin(x))^\prime=1\times\sin(x)+x\cos(x)=\sin(x)+x\cos(x)\).

Step3: Find the derivative of \(\cos(x)\) and \(C\)

The derivative of \(\cos(x)\) is \(-\sin(x)\) (since \((\cos(x))^\prime=-\sin(x)\)), and the derivative of a constant \(C\) is \(0\) (since \((C)^\prime = 0\)).

Step4: Combine the results

\(f^\prime(x)=\sin(x)+x\cos(x)-\sin(x)+0\). Simplifying the expression \(\sin(x)+x\cos(x)-\sin(x)\) gives \(x\cos(x)\).

Answer:

\(x\cos(x)\)