A calculus of the absurd

11.2 Derivatives of sums

Let’s suppose we have a function \(f(x) = q(x) + r(x)\), then the derivative of \(f(x)\) is 7272 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}