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))