Derivative of a linear combination of sine and cosine

Differentiate each term separately using the sum/difference rule: $f'(x) = \frac{d}{dx}[5\sin(x)] - \frac{d}{dx}[3\cos(x)]$. By the constant multiple rule, this becomes $5\cdot\frac{d}{dx}[\sin(x)] - 3\cdot\frac{d}{dx}[\cos(x)]$.

Recall that $\frac{d}{dx}[\sin x] = \cos x$ and $\frac{d}{dx}[\cos x] = -\sin x$. Substituting these gives $f'(x) = 5\cos(x) - 3(-\sin(x)) = 5\cos(x) + 3\sin(x)$.