A calculus of the absurd

10.2 Derivatives of sums

Let’s suppose we have a function \(f(x) = q(x) + r(x)\), then the derivative of \(f(x)\) is ⁶⁶⁶⁶ Note that this relies on the property that the limit of two things added together is the same as the sum of the limits of the two things
\[ \lim _{x \to a} (z(x) + q(x)) = \lim _{x \to a} z(x) + \lim _{x \to a} q(x) \]
Where \(z(x)\) and \(q(x)\) are any functions of \(x\) whose limit is defined as \(x \to a\).

\begin{align} \frac {df}{dx} &= \lim _{h \to 0} \frac {f(x+h) - f(x)}{h} \\ &= \lim _{h \to 0} \frac {q(x+h) + r(x+h) - q(x) - r(x)}{h} \\ &= \lim _{h \to 0} \frac {q(x+h) - q(x) + r(x+h) - r(x)}{h} \\ &= \lim _{h \to 0} \frac {q(x+h)-q(x)}{h} + \lim _{h \to 0} \frac {r(x+h)-r(x)}{h} \\ &= \frac {dq}{dx} + \frac {dr}{dx} \end{align}

That is to say that

\begin{equation} \label {linearity of differentiation} \frac {d}{dx}(a(x) + b(x)) = \frac {d}{dx} (a) + \frac {d}{dx} (b) \end{equation}