Before we can talk about "the derivative of the derivative", every single symbol on the parent page must be earned. Below we build them one at a time, each on top of the last, from a smart-12-year-old starting point.
Picture (figure below): the parabola f(x)=x2 drawn on axes. A yellow dot sits at input x=1.5 on the horizontal axis; a dashed yellow line runs up to the curve and across to the vertical axis, showing the matching output f(x)=2.25. This makes concrete the idea "one input down here → one output over there".
Why the topic needs it: everything here starts with a function. Position over time, height of a curve — all are functions. You cannot differentiate what you cannot first name as a rule.
Picture (figure below): a right triangle drawn on the curve. Its green horizontal side is the runh; its red vertical side is the risef(x+h)−f(x); the yellow line joining the two curve points is the slope. Slope = rise ÷ run, read straight off the triangle.
Why "rise over run"? Because it answers "for every unit I move right, how many units do I climb?" — a pure measure of steepness, independent of where you started.
The slope in §2 is between two points that are h apart. But we want the steepness at one exact point. So we shrink h down toward 0 and watch what number the slope settles on.
Picture (figure below, left panel): three coloured lines cut through the curve at points h apart; as h shrinks (red → orange → green) they rotate toward the white tangent line that just kisses the curve at one point — its slope is the limit. The right panel shows f(x)=∣x∣: sliding in from the left gives slope −1, from the right gives +1, so at the corner the "kiss" is ambiguous and no single limit exists.
Why the topic needs it: without the limit, "slope at a single point" has no meaning. The whole idea of a derivative is built on this shrinking process. See Derivative — definition as a limit.
Now we can name the machine the parent page relies on.
Two costumes for the same idea:
Why two notations? Lagrange is compact; Leibniz reminds you what is changing with respect to what (y with respect to x), which is priceless when there are many variables.
The parent page uses time as the input. Here are those letters.
Picture (in words): imagine a car on a straight road. s = the distance mark you're next to; v = the speedometer reading; a = the push into your seat you feel when the speed itself changes. Three physical sensations, each one the rate of change of the one before it.
Why the topic needs it: these are the physical names that make higher-order derivatives feel real. Deeper treatment in Kinematics — position, velocity, acceleration.
Picture (in words): literally count the dots stacked on top of the letter — one dot per time-derivative taken. y˙ has one dot (differentiated once with respect to t); y¨ has two dots stacked above (differentiated twice). The dots are a tally mark, nothing more.
Why: physicists differentiate with respect to time constantly, so they invented the shortest possible symbol for it.
Picture (figure above): the curve f(x)=x3. The left half (x<0) is drawn red — it frowns ⌢ because f′′=6x<0 there. The right half (x>0) is drawn green — it cups ⌣ because f′′=6x>0. The yellow dot at the origin is the inflection point where the bend flips.
Why the topic needs it: this is why we bother with the second derivative — it reveals curvature the first derivative cannot. More in Concavity and the Second-Derivative Test and Inflection points.
The parent page differentiates polynomials and sinx, so those two rules appear the most — but they are not the only tools. Here is the working toolkit, so you're never ambushed.
Read the arrows as "is needed for": each box must be understood before the box it points to. Follow the single chain down the middle — function → slope → limit → derivative — then see how it branches into the two things the topic actually cares about: the physical ladder (position→velocity→acceleration) and the curvature/higher-order side. The final box is the parent topic this page prepares you for.
Test yourself. Each line below is written as prompt ::: answer: cover everything to the right of the three colons, read the prompt on the left, say your answer out loud, then uncover to check. (This ::: "reveal" format is used throughout the vault for self-testing.)
A function f(x) is
a rule that turns one input number into exactly one output number.
The slope between two points is
the change in output divided by the change in input, hf(x+h)−f(x).
limh→0 means
the value an expression approaches as h shrinks toward zero — and it exists only if the left and right approaches agree.
The derivative f′(x) is
the limit of the slope as the step h→0 — the slope function of f — when that limit exists.
A function fails to be differentiable at
corners (left/right slopes differ) and cusps (slope blows up) — the defining limit doesn't exist there.
dxdy (Leibniz) is read as
the rate of change of y with respect to x — one indivisible symbol for the slope.
The dot in y˙ means
one derivative with respect to time; y¨ means two.
f(n)(x) means
f differentiated n times (parentheses = counter, not a power).
dx2d2y is NOT
(dxdy)2; the 2's count operator applications, giving 6x vs 9x4 for x3.
f′′>0 on an interval means the graph is
concave up, cup-shaped ⌣ (slope increasing).
f′′<0 means the graph is
concave down, frown-shaped ⌢ (slope decreasing).
An inflection point is
a point where f′′changes sign; f′′=0 alone is not enough (e.g. x4 at 0).
The power rule dxdxn=nxn−1 needs n to be
a fixed constant exponent — it does NOT apply to variable exponents like 2x.
