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Derivatives — study guide
The same fragments the interactive model serves, read in order. One source, two views.
The slope of a curve, at one exact point
A straight line has one slope, the same number wherever you measure it. A curve does not — it bends, so "how steep is this curve" only means something once you also say where. The derivative answers exactly that question: the instantaneous rate of change of a function at a single point, the slope the curve has right there and nowhere else.
The figure on this page shows the picture that matches the idea: a smooth curve with one straight line touching it at a single point, tilted to match the curve's own direction there. That line is the tangent line, and its slope is the derivative's value at that point — for f(x) = x^2 at x = 1, the tangent line has slope 2, no more and no less.
A curve built from sums of whole-number powers, or one that is already a straight line, has a derivative at every point where the question makes sense to ask. The rest of this concept is about pinning that idea down precisely enough to compute it exactly, rather than just eyeballing it from a picture.
From a chord across the curve to a line touching it
Start with two points on the curve instead of one: the point you care about, at x, and a second point a short distance away, at x + h. The straight line through them is a secant line, and its slope is easy to compute exactly — the same rise-over-run slope formula that describes a straight line, applied here to two points that happen to sit on a curve: [f(x + h) - f(x)] / h.
Now shrink h. The second point slides back along the curve toward the first, the secant line pivots to follow it, and its slope changes with it — the same move as zooming in on the curve near that point until it looks straight, like zooming in on a curved road on a map until the stretch you can see looks perfectly straight. On f(x) = x^2 near x = 1, the secant slope works out to 2 + h exactly: at h = 2 the slope is 4, at h = 1 it is 3, at h = 0.5 it is 2.5, and the figure on this page draws all three, rotating steadily toward one final position.
That final position is the tangent line, and its slope is what the secant slope is heading toward as h shrinks toward 0 — a question about what a value approaches, not a division-by-zero problem, since h never actually reaches 0. That limiting slope is the formal definition of the derivative, written precisely as the limit of that secant slope as h approaches 0.
The difference quotient, and its limit
Write the secant slope from x to x + h as a single expression, [f(x + h) - f(x)] / h, and give it a name: the difference quotient. It is defined for every h except h = 0, where it would divide by nothing, and its value is exactly the number a secant line through those two points would have.
The derivative of f at x, written f'(x), is the limit of the difference quotient as h approaches 0: f'(x) = lim_{h -> 0} [f(x + h) - f(x)] / h. That limit has to exist for the derivative to exist — approaching from positive h and from negative h must reach the same number, the same requirement every limit already carries, not a new rule invented just for derivatives.
When the limit does exist, f is called differentiable at that point, and the single number it settles on is the tangent line's slope — a curve's tangent line and its derivative are two names for the same underlying number. When the limit does not exist, there is simply no derivative there, which is not a rare edge case: it happens at any point where the curve has a sharp corner instead of a smooth turn.
Differentiating a polynomial term by term
Computing a derivative straight from the limit works, but it is slow, and a sum of powers has a shortcut. The power rule says that for any real exponent n, d/dx(x^n) = n * x^(n-1): multiply by the old exponent, then drop the exponent by one. For f(x) = x^3, that gives f'(x) = 3x^2, and for f(x) = x^2 it gives f'(x) = 2x — the slope formula behind the tangent lines this concept keeps drawing.
Two more rules let the power rule handle a whole polynomial, not just one term. The constant-multiple rule says a coefficient rides along unchanged: d/dx(5x^2) = 5 * d/dx(x^2) = 10x. The sum rule says a polynomial's derivative is the sum of its terms' derivatives, and the derivative of a plain constant is 0, since a constant's graph is a flat line with zero slope everywhere.
Put all three together on f(x) = 5x^2 - 3x + 7 and every term differentiates on its own: 10x from the first term, -3 from the second, 0 from the constant, for a total of f'(x) = 10x - 3. The same three rules handle any polynomial, one term at a time, with no limit computation required once they are established.
Differentiable implies continuous — not the other way around
A function is differentiable at a point when its derivative exists there — the difference-quotient limit settles on one number. That turns out to be a strong condition: if f is differentiable at a point, it is automatically continuous there too, meaning its graph has no break, jump, or hole at that point. A curve cannot have a well-defined tangent line at a point where the graph itself is torn.
The reverse implication fails, and the standard example is f(x) = |x| at x = 0. The graph is unbroken there — like a crease in a folded sheet: nothing is torn, but two different directions meet at the fold with no single direction between them — approaching from the left and from the right both land on f(0) = 0, so the function is continuous. But the slope approaching from the right is +1 and the slope approaching from the left is -1; those two one-sided slopes disagree, the difference-quotient limit does not settle on one number, and the corner in the figure on this page is exactly where that disagreement shows up.
So continuity is necessary for differentiability but not sufficient: every differentiable function is continuous, but plenty of continuous functions have corners, cusps, or other points where no single tangent line fits. The condition to check is always the limit itself, not whether the graph merely looks unbroken.
Writing the derivative — f'(x) versus dy/dx
Two notations for the same quantity show up constantly, and neither one is more correct than the other. Lagrange notation writes the derivative of f as f'(x), a prime mark after the function name, and it reads naturally as a new function, built from f, that reports the slope at each x. Joseph-Louis Lagrange introduced this prime notation later in the eighteenth century — an early precursor form around the 1770s, with the familiar f'(x) written out in his 1797 treatise on the theory of analytic functions.
Leibniz notation instead writes dy/dx, treating the derivative as a ratio of two infinitesimally small changes: dy, the tiny change in the output, over dx, the tiny change in the input that caused it. Gottfried Wilhelm Leibniz introduced the differential notation this form descends from in the 1670s and 80s — a manuscript in 1675, published formally in 1684 — and the ratio framing is what makes dy/dx read naturally once several changing quantities have to be tracked against each other at once.
Both notations name the exact same number at a given point; the choice is about which framing helps more in context. f'(x) emphasizes the derivative as a function in its own right, one that can be evaluated at different x values the way f itself can. dy/dx emphasizes the derivative as a rate, one quantity's change measured against another's, which is why it is the notation of choice once rates of change become the point.
What the sign of the derivative says about the curve
The sign of f'(x) reads directly off the shape of the curve at that point, without doing any further arithmetic. A positive derivative means the tangent line tilts upward, and the curve is rising there — increasing x a little increases f(x) too. A negative derivative means the tangent line tilts downward, and the curve is falling. A derivative of exactly 0 means the tangent line is level: the curve is momentarily flat, neither rising nor falling at that instant.
f(x) = x^3 - 3x shows all three in one curve, and its derivative, f'(x) = 3x^2 - 3, makes the reading exact rather than approximate. At x = -2, f'(-2) = 9, positive, and the figure on this page draws the tangent line there tilted up. At x = -1, f'(-1) = 0, and the tangent line is level — the curve has stopped rising and is about to fall. At x = 0, f'(0) = -3, negative, and the tangent line tilts down.
A flat tangent line like the one at x = -1 marks a point worth a second look, since the curve changes direction on either side of it — that single observation is the seed of an entire technique for finding a curve's highest and lowest points, one this concept only gestures at rather than works out in full.
Where an instantaneous rate shows up next
Anything that has to react to how fast something is changing right now, not on average, needs a derivative underneath it — a car correcting its own steering as its heading drifts, or a training loop adjusting one number to shrink its error, both act on an instantaneous rate rather than on how much things changed a while ago.
Both are instances of the same idea this concept has built from the ground up: a function's rate of change at one point, defined as the limit of a shrinking secant slope, computable exactly for a sum of powers with three rules, and readable at a glance from its sign. Putting that idea to work on steering, on learning, or on anything else never changes the definition — it only changes what the function stands for.
What changes next is the question asked of it: treating the input as time, holding some inputs steady while one moves, or building the number back up from small pieces instead of tearing it down into them. Every one of those questions starts from exactly the number defined here.