Intuition The one core ideaRight next to a point, a smooth curve and its straight tangent line are almost the same thing — so you can trade hard curve-math for easy line-math for a short distance. Everything on this page is just the vocabulary (points, slopes, changes, limits) you need before that sentence makes sense.
This page assumes you have seen none of the notation in Linear approximation and differentials . We build every symbol from the ground up, in the order the topic actually needs them.
f ( x ) function is a machine: you feed it a number x f ( x ) f x
Picture it as a curve on a grid. The horizontal axis is the input x f ( x ) x f ( x )
Worked example What figure 1 showsThe cyan curve is f ( x ) = x is the output f ( x )
Intuition Why the topic needs thisLinear approximation is entirely about one curve near one input . If you can't picture "curve = all the output-heights above the inputs," none of the later symbols land.
a a chosen fixed input — one particular spot on the horizontal axis where we plant our feet. It is a number, not a variable; we pick it and keep it fixed.
f ( a ) a f ( a ) height of the curve exactly above a . The dot ( a , f ( a )) on the curve.
Intuition Why a fixed base point?The whole trick is "start from a place I already know exactly." a f ( a ) 4 = 2
Before slope of a curve , get slope of a straight line .
Definition Slope of a lineSlope = rise over run . Pick two points on the line. Run is how far you move sideways (change in x Rise is how far you move up (change in output). Slope = run rise
A slope of 2 1 2 0
Worked example What figure 2 showsThe amber horizontal bar is the run (how far you move across); the white vertical bar is the rise (how far you move up over that run). The line's slope is the white bar divided by the amber bar — here 1.8 ÷ 3 = 0.6
Common mistake Not every tangent has a finite slopeRise-over-run assumes the line isn't perfectly vertical. Some curves have a vertical tangent at a point — the tangent points straight up, so the "run" is 0 0 rise f ( x ) = 3 x x = 0 finite slope, so watch for these spots — they are the one place the whole method breaks down.
Intuition Why we need slopeThe tangent line is defined by where it starts and which way it points . "Which way it points" is exactly its slope. Without slope there is no direction to walk.
A curve bends, so it has no single slope. But you can measure the slope of the straight line joining two points on it.
Definition Difference quotientTake our base input a x ( a , f ( a )) ( x , f ( x )) secant line ) has slope
x − a f ( x ) − f ( a ) .
Worked example What figure 3 showsTwo amber dots sit on the curve: the fixed base a x secant . The white bars are its run (x − a f ( x ) − f ( a ) a
Intuition Why this shape of fraction?It is the only honest way to say "average steepness of the curve between a x this tool because slope means rise-over-run, and these are the two heights and two positions we actually have.
lim x → a ( something with x ) something gets closer and closer to as x a x a x → a x a
Definition One-sided limits and where they must existx a two directions : from the right (inputs bigger than a x → a + a x → a − lim x → a only exists when both sides agree on the same value. If the function isn't even defined on one side — for example x 0 lim x → 0 x a both sides first.
Intuition Why a limit and not just "plug inx = a x = a 0 0 approaches as the secant's two points slide together, without ever colliding. That approached number is the slope of the curve itself.
The tool exists precisely to turn a secant (two points) into a tangent (one point) by a controlled squeeze. See Derivative as a limit .
Definition Derivative at a pointf ′ ( a ) = lim x → a x − a f ( x ) − f ( a ) . ′ f ′ ( a ) the slope of the curve exactly at a — the slope of the tangent line that just kisses the curve there (provided that limit exists and is finite; recall §3's vertical-tangent warning).
Worked example What figure 4 showsThe two faint white lines are secants for inputs x a x a amber tangent line — whose slope is exactly f ′ ( a ) is the limit of §5 happening.
f ′ a any input where the tangent slope exists and you get a slope there too. So f ′ machine in its own right : feed it an input, it returns the slope of f f ′ ( a ) f ′ ( x )
Intuition Why the topic lives or dies on thisLinear approximation is "start at f ( a ) f ′ ( a ) is that pointing direction. This is the single most important symbol on the page.
Definition The output namey f ( x ) y = f ( x ) y is the height of the curve above x y f ( x ) y y vertical axis .
Now let's line up the input-symbols so nobody gets lost. Up to now a fixed base and x nearby input. In this section we care about the step from a starting input to a slightly moved input, so we rename things to spotlight that step:
a x Δ x d x
Start at some input (call it x a
Move it by a small amount. That move is the step .
Δ x true, actual change: Δ x = ( new input ) − ( old input ) ( x − a ) d x the exact same step , but the name we use when we're going to ride the tangent line with it. In this whole topic d x Δ x same chosen number ; we only keep two names to flag "true-curve change" vs "tangent-line change." So d x = Δ x
Definition The change symbolΔ Δ actual change in . So Δ x y = f ( x ) Δ y = f ( x + Δ x ) − f ( x ) true output height y actually moved as a result, measured along the curve .
Definition The differentialsd x d y d x chosen small step in the input (equal to Δ x d y y if you stayed on the straight tangent line for that step:
d y = f ′ ( x ) d x = ( slope at x ) × ( step ) . f ′ ( x ) x
Δ y d y y Δ y curve (bends away); d y straight tangent (goes straight). Even though d x = Δ x outputs differ: they agree only for tiny steps — the gap between them is the approximation error. Never write Δ y = d y Δ y ≈ d y
Intuition Why we need bothΔ y d y Δ y ≈ d y
≈ a ≈ b a approximately equal to b "...and the smaller the step, the truer this gets."
Now every piece of L ( x ) = f ( a ) + f ′ ( a ) ( x − a )
The figure below is a study checklist made visual : each box is one section above, and each arrow means "you need the box behind the arrow before the box in front makes sense." Trace it from top to bottom to confirm you've met every prerequisite before tackling Linear approximation and differentials — if any box feels shaky, reread that section.
Each line below is a mini flashcard: the part before the ::: is the question, the part after it is the answer. Cover the right side, answer out loud, then reveal to check yourself.
What does f ( x ) The output a function-machine gives when fed input x x
What is a f ( a ) a f ( a )
State "slope" without any formula, then with one. Rise over run — up-distance divided by across-distance; slope = run rise
When does slope (rise over run) fail to give a number? At a vertical tangent — the run is 0
Write the difference quotient and say what it measures. x − a f ( x ) − f ( a ) ( a , f ( a )) ( x , f ( x ))
Why can't you just plug x = a It gives 0 0 x → a
What must be true for a two-sided limit lim x → a The left-hand and right-hand limits must exist AND agree; a
Define f ′ f ′ ( a ) The slope machine: a function that returns the tangent slope of f
What does y = f ( x ) y
Is d x Δ x Yes — same chosen input step; the two names only flag "tangent-line change" vs "true-curve change" in the output.
Distinguish Δ y d y Δ y y d y = f ′ ( x ) d x y
What does ≈ Approximate equality that gets truer the smaller the step becomes.
Read L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) Start + Slope × Step.
Derivative as a limit — §5 and §6 are its whole content; this page is the on-ramp.Tangent line — the geometric object L ( x ) Linear approximation and differentials — the parent topic every symbol here feeds. 
