Derivatives

The motivation (why we need a new idea)

You already know average rate of change: between two points $x=a$ and $x=b$, the slope is

That’s a two-point measurement. But many real questions are one-point questions:

• •“How fast is the position changing right now?” (instantaneous velocity)
• •“How steep is the curve at this point?” (tangent slope)
• •“If I nudge $x$ a tiny bit, how much does $f(x)$ change?” (sensitivity)

To answer those, we take the average rate of change over a smaller and smaller interval and ask whether it approaches a limiting value.

Definition (difference quotient → limit)

Fix a point $a$. Consider a nearby point $a+h$ (so $h$ is a small horizontal step). The slope of the secant line through

• •$P=(a, f(a))$ and
• •$Q=(a+h, f(a+h))$

is

If this slope approaches a single number as $h \to 0$, we define the derivative:

This is the defining limit (also called the definition via the difference quotient).

Two interpretations you should hold at the same time

1) Geometric: $f'(a)$ is the slope of the tangent line to the graph at $x=a$.

2) Dynamic (rate): $f'(a)$ is the instantaneous rate of change of $f$ with respect to $x$ at $x=a$.

Both interpretations come from the same limit.

Units (a quick reality check)

If $x$ is measured in seconds and $f(x)$ in meters, then

• •$\Delta x$ has units seconds
• •$\Delta f$ has units meters
• •$\frac{\Delta f}{\Delta x}$ has units meters/second

So $f'(a)$ has the same units as a speed. This “units check” is one of the simplest ways to catch mistakes.

Supporting static diagram (to reduce cognitive load)

Static diagram to include in the lesson UI: a coordinate plane with the curve $y=f(x)$, points $P=(a,f(a))$ and $Q=(a+h,f(a+h))$, the secant line through $P$ and $Q$, and labels for:

• •horizontal change: $h$
• •vertical change: $f(a+h)-f(a)$
• •secant slope: $\frac{f(a+h)-f(a)}{h}$

This diagram should remain visible while the definition is introduced so the symbols feel anchored to geometry.

What the interactive canvas should show (visual layer)

Your interactive canvas should make the limit feel physical.

Canvas scene A: Secant-to-tangent animation

• •Plot a smooth function, e.g. $f(x)=x^2$.
• •Show point $P$ fixed at $x=a$.
• •Show point $Q$ at $x=a+h$ with a slider for $h$.
• •Draw the secant line $\overline{PQ}$.
• •Display the numeric value of $m_{\text{sec}}(h)$.
• •As the learner drags $h \to 0$, animate $Q \to P$ and show the secant line rotate toward the tangent line.

Canvas scene B: One-sided convergence

• •Add a toggle: “Approach from the right ($h\to 0^+$)” vs “Approach from the left ($h\to 0^-$)”.
• •Show two slopes: $m_{\text{sec}}(h>0)$ and $m_{\text{sec}}(h<0)$.
• •Emphasize: differentiable requires both sides approach the same value.

Canvas scene C: Non-differentiable corner/cusp toggle

• •Toggle function between:
• •$f(x)=|x|$ (corner at 0)
• •$f(x)=\sqrt[3]{x}$ (vertical tangent at 0)
• •Keep the same secant-to-tangent mechanics.
• •Show that left and right slopes disagree for $|x|$ and that slopes blow up for $\sqrt[3]{x}$.

These visuals directly support the three atomic concepts: instantaneous rate, defining limit, and tangent slope.

Start with what you know: secant slope

Pick two x-values: $a$ and $a+h$.

The secant slope is

This is an average rate of change on the interval $[a, a+h]$.

Shrink the interval (the key move)

To get a one-point notion, we shrink the interval by taking $h$ smaller and smaller.

• •If $h=1$, you’re measuring across a wide interval.
• •If $h=0.1$, you’re measuring locally.
• •If $h=0.001$, you’re extremely local.

The derivative is the number the secant slopes are heading toward—if they head toward a single number at all.

Why the limit is necessary (not optional)

You might wonder: why not just set $h=0$ in

Because you would get

and $\frac{0}{0}$ is indeterminate (it doesn’t have a single value). The limit process asks: as we approach the problematic point, does the expression stabilize?

One-sided limits and differentiability

A subtle but crucial point: $h \to 0$ means approaching 0 from both sides.

• •Right-hand derivative:

• •Left-hand derivative:

A function is differentiable at $a$ only if both exist and are equal:

This is exactly what your “one-sided slope convergence” canvas should make obvious: you can see when the two sides don’t agree.

Tangent line equation (once you have the slope)

If you know $f'(a)$, you know the slope of the tangent line at $x=a$. The tangent line passes through $(a, f(a))$ with slope $f'(a)$:

This is often the fastest way to turn “derivative” into a concrete geometric object.

Micro-change viewpoint (a preview of linear approximation)

When $h$ is small, the change in $f$ is approximately

This is the intuition behind many applications: derivatives translate a tiny input change into an approximate output change.

What the interactive canvas should emphasize here

To reinforce the limit idea (instead of treating it as symbolism):

• •Display a table that updates live as $h$ changes:

• •The learner should notice convergence: the slope values settle.
• •When switching to a corner/cusp example, the table should fail to settle or show two different settling values from left vs right.

Differentiable vs. continuous (don’t merge these)

A common mental trap is to assume:

• •“If the graph has no jumps, it must have a derivative.”

Not always.

Facts to keep straight:

• •If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
• •But $f$ can be continuous at $a$ and still not differentiable there.

So differentiability is a stronger condition.

Three classic ways differentiability can fail

1) Corner (left and right slopes disagree)

Example: $f(x)=|x|$ at $x=0$.

• •For $x>0$, the slope is like the line $y=x$ (slope 1).
• •For $x<0$, the slope is like the line $y=-x$ (slope −1).

Compute using the definition:

For $h>0$:

For $h<0$:

So

They differ, so $f'(0)$ does not exist.

Interactive tie-in: in your corner toggle, the secant line will approach two different tangent candidates depending on direction.

2) Cusp or vertical tangent (slope blows up)

Example: $f(x)=\sqrt[3]{x}$ at $x=0$.

Consider the difference quotient:

As $h\to 0$, $h^{-2/3}\to \infty$. The slope becomes unbounded (vertical tangent). In many calculus courses, we say the derivative does not exist as a finite number.

Interactive tie-in: your slope readout should grow very large in magnitude as $h$ shrinks.

3) Discontinuity (jumps or holes)

If $f$ isn’t continuous at $a$, it cannot be differentiable there.

This is less subtle visually—your curve breaks—so it’s a good “sanity check” case.

Differentiability is local smoothness

A useful phrase: differentiable at $a$ means the graph is locally well-approximated by a line near $a$.

That’s why the “zoom in, it becomes a line” intuition works. A corner never becomes a single line no matter how much you zoom; it keeps its sharpness.

A quick comparison table

What the canvas should show in this section

To make “failure modes” memorable:

• •Keep the same point $P$ and slider for $h$.
• •Let learners switch functions with a toggle.
• •Show both one-sided secant slopes simultaneously (color-code left vs right).
• •Add a small indicator:
• •“Differentiable here: yes/no”
• •If no, show reason: “left/right mismatch” or “slope unbounded”.

Derivatives as a sensitivity tool

At a point $a$, the derivative tells you how sensitive the output is to tiny input changes.

If $\Delta x$ is small, then

This is the bridge from geometry to practical estimation.

Example intuition: If a cost function has derivative 5 dollars/unit at the current production level, then producing one more unit increases cost by about $5.

Derivatives as slope fields for behavior

Even before learning derivative “rules,” knowing that $f'(x)$ is slope gives a powerful way to interpret a graph:

• •Where $f'(x)>0$, the function is increasing.
• •Where $f'(x)<0$, the function is decreasing.
• •Where $f'(x)=0$, the tangent is horizontal (possible maxima/minima).

This connects directly to optimization ideas used everywhere (including machine learning).

Why this node unlocks the next nodes

Link to Derivative Rules

The definition is conceptually perfect but computationally slow. The next node (Derivative Rules) gives shortcuts (power/product/quotient/chain rules). Those rules are justified because they match what the limit definition produces.

Link to Maximum Likelihood Estimation (MLE)

In MLE, you maximize a likelihood (or log-likelihood) with respect to parameters. “Maximize” often means “take derivative, set it to zero, solve.” The derivative is what turns “best parameter” into an equation you can solve.

Link to Convex Functions

Convexity uses derivatives to formalize “curves that bend upward.” For differentiable functions, one hallmark is that the derivative is increasing. So understanding $f'(x)$ as slope makes convexity feel natural.

Link to Integrals

Integrals and derivatives are paired by the Fundamental Theorem of Calculus. Informally: differentiation measures instantaneous change, integration accumulates change. This node provides the “change” half of that story.

A small, concrete bridge: tangent line approximation

Once you can compute $f'(a)$, you can approximate $f$ near $a$ by its tangent line:

This idea appears constantly later: numerical methods, error estimates, optimization steps, and more.