Question

let ( f(x)=3sin(x)+11cos(x) ). find the following:

1. ( f(x)=)
2. ( f(3)=)

Explanation:

Step1: Differentiate \(f(x)\)

Use the derivative rules: \((\sin x)'=\cos x\) and \((\cos x)'=-\sin x\).
For \(f(x) = 3\sin(x)+11\cos(x)\), by the sum - rule \((u + v)'=u'+v'\) where \(u = 3\sin(x)\) and \(v = 11\cos(x)\).
The derivative of \(u\): \((3\sin(x))'=3\cos(x)\) (using the constant - multiple rule \((cf(x))'=cf'(x)\) with \(c = 3\) and \((\sin x)'=\cos x\)).
The derivative of \(v\): \((11\cos(x))'=- 11\sin(x)\) (using the constant - multiple rule \((cf(x))'=cf'(x)\) with \(c = 11\) and \((\cos x)'=-\sin x\)).
So, \(f'(x)=3\cos(x)-11\sin(x)\).

Step2: Evaluate \(f'(3)\)

Substitute \(x = 3\) into \(f'(x)\).
\(f'(3)=3\cos(3)-11\sin(3)\).

Answer:

1. \(f'(x)=3\cos(x)-11\sin(x)\)
2. \(f'(3)=3\cos(3)-11\sin(3)\)