Question

calculate the derivative, if ( f(x)=-25xcos(x) )
( f(x)=-25xsin(x)+25cos(x) )
( f(x)=25xsin(x)-25cos(x) )
( f(x)=-25xsin(x)-25cos(x) )

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = uv\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u=-25x\) and \(v = \cos(x)\). First, find \(u^\prime\) and \(v^\prime\). \(u^\prime=-25\) (since the derivative of \(ax\) with \(a = - 25\) is \(a\)), and \(v^\prime=-\sin(x)\) (derivative of \(\cos(x)\) is \(-\sin(x)\)).

Step2: Substitute into the product rule

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Answer:

B. \(f^\prime(x)=25x\sin(x)-25\cos(x)\)