Introduction
Purpose of this Series
In this series, I want to prove all of the main single-variable derivative relations from calculus. This has been done a thousand times in a thousand different ways, so I don’t think I’m adding anything to the space. I guess this is mainly just for my own enjoyment.
I am going to keep the proofs at the level of first or second-year calculus. I don’t want to get into Cauchy series, continuity, and delta-epsilon definitions. I have other series that dive into these topics [1, 2].
Intuition of a Derivative
There is a precise delta-epsilon definition of a derivative, but again I don’t want to get into that stuff in this series. Thus, I am going to just assert the limit definition and give a general intuition.
Often we are interested in the rate of change of a function. The meaning of this is exactly as the name would imply. It is a measure of how much the function is changing over some interval. Suppose \(f\) is a function. Then the average rate of change between two points $a$ and $b$ is
\[\frac{\Delta f}{\Delta x} = \frac{f(b) - f(a)}{b - a}\]Geometrically, the average rate of change is the slope of the secant line through $f$ from $a$ and $b$.
We want to define a notion of instantaneous rate of change, i.e. the amount a function is changing at a single point. On the surface, this doesn’t make any sense. How can we define a change in $f$ if we don’t have a change in $x$? We can imagine the coordinate $(b, f(b))$ sliding down the curve towards $(a, f(a))$. As it does so, the secant line approaches a line that just touches the function as a single point. This is called a tangent line. The slope of the tangent line is our instantaneous rate of change.
Intuitively, what we are saying is that if we zoom far enough into the point $(a, f(a))$, then the function $f$ will look like a straight line, which has a constant rate of change. Thus, the instantaneous rate of change is well-defined.
The Definition of a Derivative
We can rigorously define this notation of “zooming in far enough” by considering a point very, very close to $(a, f(a))$. As we get closer and closer to $(a, f(a))$, the secant line will converge to a tangent line. That is the slope we are interested in. In other words, we want the limit as two points of a secant line become one point of a tangent line. There are two equivalent ways to define this.
\[\frac{df}{dx}\bigg\rvert_{x=a} = f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a} \qquad\qquad\qquad \frac{df}{dx} = f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\]Notice the different notations. I will be using both throughout the series.
In the left definition, we fix coordinate $(a, f(a))$ and we consider the secant line from that point to the point $(x, f(x))$. Then, we take the limit as $(x, f(x))$ comes closer and closer to $(a, f(a))$. In the right definition, we fix coordinate $(x, f(x))$. Then, we consider another coordinate a small distance away $(x+h, f(x+h))$ and the corresponding secant line. We see what happens as $h$ approaches $0$, and thus the secant line becomes a tangent line.
In practice, it turns out the right definition is easier to compute since we’ve decoupled the limit from the particular coordinate. Also, the limit of things approaching $0$ is simpler than approaching some arbitrary number $a$.
Limit Laws
Limits have a precise and technical definition. For this series, I will just assert these properties of limits that we will use. I prove these rigorously in my series on limits and continuity. I list the relevant laws here.
Let $f$ and $g$ be functions. Let $c \in \mathbb{R}$ be any constant.
\[\lim_{x \rightarrow a} (c \cdot f(x)) = c \cdot \left ( \lim_{x \rightarrow a} f(x) \right ) \\[10pt] \lim_{x \rightarrow a} (f(x) + g(x)) = \left ( \lim_{x \rightarrow a} f(x) \right ) + \left ( \lim_{x \rightarrow a} g(x) \right ) \\[10pt] \lim_{x \rightarrow a} (f(x) - g(x)) = \left ( \lim_{x \rightarrow a} f(x) \right ) - \left ( \lim_{x \rightarrow a} g(x) \right ) \\[10pt] \lim_{x \rightarrow a} (f(x) \cdot g(x)) = \left ( \lim_{x \rightarrow a} f(x) \right ) \cdot \left ( \lim_{x \rightarrow a} g(x) \right ) \\[10pt]\]Suppose $\lim_{x \rightarrow a} g(x) \neq 0$, then
\[\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{ \lim_{x \rightarrow a} f(x) }{ \lim_{x \rightarrow a} g(x) } \\[10pt]\]Let $n, m \in \mathbb{N}$ such that $m > 0$. If $m$ is even, then assume $\lim_{x \rightarrow a} f(x) > 0$.
\[\lim_{x \rightarrow a} (f(x))^{n/m} = \left ( \lim_{x \rightarrow a} f(x) \right )^{n/m}\]If $f$ is continuous, then
\[\lim_{x \rightarrow a} f(x) = f(a) \\[10pt] \lim_{x \rightarrow a} f(g(x)) = f \left ( \lim_{x \rightarrow a} g(x) \right ) \\[10pt]\]If both $f$ and $g$ are continuous, then
\[\lim_{x \rightarrow a} f(g(x)) = \lim_{g(x) \rightarrow g(a)} f(g(x))\]The above is a bit of an abuse of notation, see the limits and continuity series for the rigorous version.
Suppose $f$ is a continuous two-variable function, then
\[\lim_{x \rightarrow a} \lim_{y \rightarrow b} f(x, y) = \lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x, y)\]