Question

if ( f(x)=2 sin x + 6 cos x ), then ( f^{prime}(x)= ) ( f^{prime}(4)= ) question help: message instructor

Explanation:

Step1: Differentiate term - by - term

Use the derivative rules: \((\sin x)^\prime=\cos x\) and \((\cos x)^\prime =-\sin x\).
For \(y = 2\sin x+6\cos x\), by the sum rule \((u + v)^\prime=u^\prime + v^\prime\) (where \(u = 2\sin x\) and \(v = 6\cos x\)), we have \(y^\prime=(2\sin x)^\prime+(6\cos x)^\prime\).
Since \((a\cdot f(x))^\prime=a\cdot f^\prime(x)\) (where \(a\) is a constant), \((2\sin x)^\prime=2\cos x\) and \((6\cos x)^\prime=- 6\sin x\). So \(f^\prime(x)=2\cos x-6\sin x\).

Step2: Evaluate \(f^\prime(x)\) at \(x = 4\)

Substitute \(x = 4\) into \(f^\prime(x)\).
\(f^\prime(4)=2\cos(4)-6\sin(4)\)

Answer:

\(f^\prime(x)=2\cos x - 6\sin x\)
\(f^\prime(4)=2\cos(4)-6\sin(4)\)