Question

find the derivative of the following function.
y = 3 sin x + 8 cos x
\frac{dy}{dx}=\square

Explanation:

Step1: Differentiate \(3\sin x\)

The derivative of \(\sin x\) is \(\cos x\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\), for \(y_1 = 3\sin x\), we have \(y_1^\prime=3\cos x\).

Step2: Differentiate \(8\cos x\)

The derivative of \(\cos x\) is \(-\sin x\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\), for \(y_2 = 8\cos x\), we have \(y_2^\prime=- 8\sin x\).

Step3: Use the sum rule

If \(y = y_1 + y_2\), then \(y^\prime=y_1^\prime + y_2^\prime\). Substituting \(y_1^\prime\) and \(y_2^\prime\) into the sum rule formula, we get \(\frac{dy}{dx}=3\cos x-8\sin x\).

Answer:

\(3\cos x - 8\sin x\)