Theorem 2 — Derivative of Difference of Functions

Theorem 2. If $u$ and $v$ are differentiable functions of $x$ such that $y=u-v$, then

The book leaves this proof as an exercise, since it is identical in structure to the sum-rule proof of Theorem 1: give $x$ an increment $\delta x$, note $\delta y=\delta u-\delta v$, divide by $\delta x$ and take the limit as $\delta x\to0$, using the differentiability of $u$ and $v$ exactly as before.

Corollary (linear combination rule). If $u,v,w,\ldots$ are any finite number of differentiable functions of $x$, and $k_1,k_2,k_3,\ldots$ are constants, then for

$\dfrac{dy}{dx}=k_1\dfrac{du}{dx}\pm k_2\dfrac{dv}{dx}\pm k_3\dfrac{dw}{dx}\pm\cdots$ …