Derivative of a linear combination of sine and cosine

Differentiate term by term using the difference rule: $h'(x) = \frac{d}{dx}[7\sin(x)] - \frac{d}{dx}[3\cos(x)]$. By the constant multiple rule, each constant factor (7 and 3) passes through the derivative unchanged, leaving $7\cdot\frac{d}{dx}[\sin x] - 3\cdot\frac{d}{dx}[\cos x]$.

Now substitute the known trig derivatives: $\frac{d}{dx}[\sin x] = \cos x$ and $\frac{d}{dx}[\cos x] = -\sin x$. This gives $h'(x) = 7\cos(x) - 3(-\sin(x)) = 7\cos(x) + 3\sin(x)$.

The sign flipped. The original term was negative three cosine of x, and since the derivative of cosine is negative sine, the negative signs multiplied together to produce a positive result. This turned negative three cosine into positive three sine in the final derivative. This flip is a direct consequence of the negative sign in the cosine derivative rule combined with the constant multiple and difference rules.