Differentiation of a sum

Let $y = u + v - w$. By the General Rule,

• FIRST STEP. $y + \Delta y = u + \Delta u + v + \Delta v - w - \Delta w$.
• SECOND STEP. $\Delta y = \Delta u + \Delta v - \Delta w$.
• THIRD STEP. $\frac{\Delta y}{\Delta x} = \frac{\Delta u}{\Delta x} + \frac{\Delta v}{\Delta x} - \frac{\Delta w}{\Delta x}$.
• FOURTH STEP. $\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} - \frac{dw}{dx}$. [Applying Theorem 3.8.1]

Therefore, $\frac{d}{dx}(u + v - w) = \frac{du}{dx}\ +\ \frac{dv}{dx}\ -\ \frac{dw}{dx}$ (equation (III) above). Similarly, for the algebraic sum of any finite number of functions.

The derivative of the algebraic sum of a finite number of functions is equal to the same algebraic sum of their derivatives.