Derivatives from First Principles

Every rule in Derivative Rules came from one calculation, done once, carefully. Doing it yourself is what makes the rules something you understand rather than something you were handed — and it is the only place where the derivative’s definition is visible.

The quotient before the limit

Take two points on a curve, $x$ and $x+h$. The slope of the line through them is rise over run:

$\frac{f(x+h)-f(x)}{h}$

That is an average rate of change over an interval of width $h$ — a secant slope, and completely ordinary. The whole of calculus is what happens when $h$ shrinks.

The definition

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Read it as an instruction: build the quotient, simplify until the $h$ in the denominator cancels, and only then let $h$ go to zero. The order matters. Substituting $h=0$ first gives $\frac{0}{0}$, which is not a number and not an answer.

Doing it, slowly

For $f(x)=x^2$:

$\frac{f(x+h)-f(x)}{h}=\frac{(x+h{)}^2-x^2}{h}=\frac{x^2+2xh+h^2-x^2}{h}=\frac{2xh+h^2}{h}$

Every term still has an $h$, so the fraction simplifies:

$\frac{h(2x+h)}{h}=2x+h(h\ne 0)$

Now the limit is safe, because nothing is dividing by zero any more:

$f^{\prime }(x)={\lim }_{h\to 0}(2x+h)=2x$

That is the power rule for $n=2$, derived rather than asserted. Do $f(x)=x^3$ the same way and $3x^2$ falls out; the pattern is visible after two or three, which is exactly how the rule was found.

Step	What you are doing	The trap
Expand	Substitute and expand fully	Forgetting a middle term of the binomial
Cancel	Subtract $f(x)$; every remaining term has $h$	Cancelling before subtracting
Divide	Factor $h$ out and cancel it	Writing $h=0$ at this stage
Take the limit	Let $h\to 0$ in what remains	Stopping before this step

Positive, negative, and zero — read off the numbers

Before any algebra, a table of values tells you most of what a derivative is for. Compute the average rate of change over short intervals across a function’s domain and watch the sign:

• Positive intervals: the function is increasing there.
• Negative intervals: it is decreasing.
• Zero, or a sign change: a maximum, a minimum, or a moment of levelling off.

Build that table in Using Desmos or a spreadsheet for a cubic, with $h=0.001$, and plot the results against $x$. What appears is the graph of the derivative — discovered numerically, before you can compute it symbolically. The Slope Detective is that investigation in full, and it is why the shape of $f^{\prime }$ feels familiar by the time you meet it algebraically.

Why bother, once you know the rules?

Three reasons. It is the definition, so every proof of every rule starts here. It is what a computer does when no rule applies — numerical differentiation is this quotient with a small fixed $h$. And when a question asks you to differentiate something the rules do not cover, first principles is the method that always works, slowly.

Curriculum connection

A2.3

determine the derivatives of polynomial functions by simplifying the algebraic expression $\frac{f(x+h)-f(x)}{h}$ and then taking the limit of the simplified expression as $h$ approaches zero [i.e., determining ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$]

Link to original

A2.1

determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals (e.g., by using graphing technology to examine the table of values and the slopes of tangents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima Sample problem: Given a smooth function for which the slope of the tangent is always positive, explain how you know that the function is increasing. Give an example of such a function.

Link to original

A2.2

generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, $f(x)$, for various values of $x$ (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, $f^{\prime }(x)$ or $\frac{dy}{dx}$, and make connections between the graphs of $f(x)$ and $f^{\prime }(x)$ or $y$ and $\frac{dy}{dx}$ [e.g., when $f(x)$ is linear, $f^{\prime }(x)$ is constant; when $f(x)$ is quadratic, $f^{\prime }(x)$ is linear; when $f(x)$ is cubic, $f^{\prime }(x)$ is quadratic] Sample problem: Investigate, using patterning strategies and graphing technology, relationships between the equation of a polynomial function of degree no higher than 3 and the equation of its derivative.

Link to original