A continuous graph can be drawn without lifting the pencil. A differentiable graph is also smooth: at every point it has one clear direction, its tangent. This chapter makes both ideas precise, then builds a toolkit that can find the slope of almost any function.
Walk along the graph towards x = c from each side. If both walks head for the same height and the dot at c sits there, the graph is unbroken.
f is continuous at c when the limit exists and equals the actual value f(c).
The height you head for from the left (x → c⁻) is the left-hand limit, LHL; from the right (x → c⁺), the right-hand limit, RHL. Both must exist and be equal.
f is continuous when it is continuous at every point of its domain. At an endpoint of [a, b], only the one-sided limit from inside counts.
LHL = RHL = f(c). Example: 2x + 3 at x = 1.
LHL ≠ RHL, so there is no limit. Example: [x] at every integer.
LHL = RHL ≠ f(c). Example: x³ + 3 with f(0) = 1.
A point that is not in the domain is not a discontinuity. 1/x is continuous: it is simply not defined at 0, so there is nothing to test there (Examples 9 and 12).
f(x) = 2x + 3 at x = 1.
f(x) = x² at x = 0.
f(x) = |x| at x = 0.
x³ + 3 (x ≠ 0), 1 (x = 0).
Constant f(x) = k.
Identity f(x) = x.
|x| everywhere.
x³ + x² − 1.
f(x) = 1/x, x ≠ 0.
x + 2 (x ≤ 1), x − 2 (x > 1).
x + 2 (x < 1), 0 (x = 1), x − 2 (x > 1).
x + 2 (x < 0), −x + 2 (x > 0).
x (x ≥ 0), x² (x < 0).
Every polynomial.
The greatest integer function [x].
Join continuous functions together and the result is still continuous. Two theorems say exactly when.
If f and g are continuous at c, so are their sum, difference and product, and their quotient as long as the bottom is not 0 at c.
If g is continuous at c and f is continuous at g(c), then the composite f ∘ g is continuous at c.
Polynomials, rational functions (away from zeros of the bottom), sin, cos, tan (where defined), and every combination of them.
Every rational function p/q.
sin x.
tan x = sin x / cos x.
sin(x²).
|1 − x + |x||.
Draw a chord from a point on the graph to a nearby point, then slide the nearby point closer. The slope the chord settles on is the derivative.
The slope of the chord from (c, f(c)) to the nearby point (c + h, f(c + h)), as the nearby point closes in.
f is differentiable at c when the left-hand derivative, LHD (h → 0⁻), and the right-hand derivative, RHD (h → 0⁺), are finite and equal.
If f is differentiable at a point, it is also continuous there.
The converse is false: |x| is continuous at 0, but its chords have slope −1 on the left and +1 on the right.
Rules
Standard derivatives (Table 5.3)
Every one of these comes from the chord-slope limit above; the basics show how.
Zoom in on a differentiable graph and it looks more and more like a straight line: its tangent. Zoom in on |x| at 0 and the corner never goes away.
When one function sits inside another, each layer magnifies a small change by its own rate. So the rates multiply.
For y = f(u) with u = g(x). Read it as: outer rate × inner rate.
One more layer, one more rate to multiply.
Put t = 2x + 1, so the function is t³. Its derivative is 3t² · 2 = 6(2x + 1)², the same as expanding the cube first.
How to use it
f(x) = sin(x²).
Sometimes y is tied to x by an equation that is hard to solve for y. You do not need to: differentiate both sides as they stand.
Treat y as a function of x. By the chain rule, every term containing y picks up a factor dy/dx.
No need to solve for y. The same works for relations such as y + sin y = cos x.
How to use it
x − y = π.
y + sin y = cos x.
The graph of an inverse is the mirror image of the original in the line y = x. So its slopes are the reciprocals of the original slopes.
In that range cos y = √(1 − x²) > 0, so you may divide by it. This gives the first row of the table below.
| f(x) | f′(x) | where |
|---|---|---|
| sin⁻¹ x | 1√(1 − x²) | −1 < x < 1 |
| cos⁻¹ x | − 1√(1 − x²) | −1 < x < 1 |
| tan⁻¹ x | 11 + x² | every real x |
f(x) = sin⁻¹x.
There is one special base, e ≈ 2.718, for which the slope of eˣ always equals its height. Its mirror image, log x, has slope 1/x.
Here log means the natural log, to base e. Its graph is the mirror image of eˣ in y = x.
The one base whose curve has slope equal to its height at every point.
Passes through (0, 1), always increases, hugs the x-axis far to the left, and grows faster than any power of x. Its inverse is logb x, with domain (0, ∞).
Log laws
Logs turn products into sums and powers into multiples. That is what makes the next section work.
Is x = elog x for all real x?
e−x, sin(log x), cos⁻¹(eˣ), ecos x.
Powers like xˣ, and long products and quotients, are hard to differentiate as they stand. Take logs first and they turn into sums.
The same first step works for a long product and quotient.
Then multiply back by y to get dy/dx.
How to use it
Both f(x) and u(x) must be positive (f(x) > 0, u(x) > 0) for their logs to exist. For a sum of such powers, take logs of each part separately.
√[(x − 3)(x² + 4)/(3x² + 4x + 5)].
aˣ.
xsin x.
yˣ + xʸ + xˣ = aᵇ.
When x and y both depend on a third variable t, a point moves along a curve as t changes. Its slope is vertical speed ÷ horizontal speed.
For x = f(t) and y = g(t), whenever dx/dt ≠ 0. The letter t is called the parameter.
This is the chain rule. Divide both sides by dx/dt.
The answer is usually in terms of t only (Example 32).
x = a cos θ, y = a sin θ.
x = at², y = 2at.
x = a(θ + sin θ), y = a(1 − cos θ).
x2/3 + y2/3 = a2/3.
Differentiate a second time and you learn how fast the slope itself is changing. That tells you which way the graph bends.
This is the second order derivative, also written f″(x), y″, y₂ or D²y.
y″ is the rate at which the slope changes. Positive: the slope is increasing and the graph bends up, like a cup. Negative: it bends down, like a cap.
Many exercises then check an equation linking y, y′ and y″, such as this one for y = A sin x + B cos x (Example 36).
y = x³ + tan x.
y = A sin x + B cos x.
y = 3e²ˣ + 2e³ˣ.
y = sin⁻¹x.
Rewrite first (powers instead of roots, change of base, a trig substitution), then pick the method.
√(3x + 2) + 1/√(2x² + 4); log₇(log x).
cos⁻¹(sin x), tan⁻¹(sin x/(1 + cos x)), sin⁻¹(2ˣ⁺¹/(1 + 4ˣ)).
(sin x)sin x.
y = at + 1/t, x = (t + 1/t)ᵃ.
sin²x with respect to ecos x.
This chapter's exercises have no multiple-choice questions, so these twelve check the key ideas. Tap an option to check it and see the reasoning.
f is continuous at c when limx→c f(x) = f(c); continuous when this holds at every point of its domain.
Sums, differences, products, quotients (bottom ≠ 0) and composites of continuous functions are continuous.
Every differentiable function is continuous, but not conversely (|x| at 0).
Chain rule: if f = v ∘ u and t = u(x), then df/dx = (dv/dt)(dt/dx).
Logarithmic differentiation handles [u(x)]v(x); u and f must be positive.
Inverse trig
e and log
Parametric and second order
Newton's "fluxions" and Leibniz's calculus found rates of change. Leibniz published first, in 1684, with the dy/dx notation we still use.
Euler's Introductio made eˣ and the natural logarithm central to analysis and used the letter e for the base.
They defined continuity through limits instead of intuition about unbroken curves.
He showed a function that is continuous everywhere yet differentiable nowhere, proving that continuity alone does not give smoothness.
Each example is a small animated story. Beads walk along a graph towards a point from the left and from the right. Functions split into machines that pass their rates along a chain. Equation tiles slide as you collect dy/dx. Most walkthroughs end with a graph check: the tangent is measured on the curve and compared with our formula. Press Next step (or Play). Under the controls, each step is explained in detail, with links to the exact basics it uses; after reading a basic, press ← Back to return to the same step. Many examples have a Try it slider.
A function is continuous at c when the heights approached from the left and from the right both equal f(c). Then we test whole functions: constants, x, |x|, polynomials and 1/x.
Inside each piece a standard function is continuous, so only the joins can break. The staircase [x] breaks at every integer.
Sums, products, quotients (where the bottom is not 0) and composites of continuous functions are continuous.
Multiply the rates of the layers; differentiate relations term by term; undo an inverse function and differentiate.
log only accepts positive numbers; eˣ is its own derivative and (log x)′ = 1/x.
Logs turn products into sums and bring exponents down, so hard functions become easy ones.
dy/dx = (dy/dt) ÷ (dx/dt): the vertical speed over the horizontal speed.
Differentiate twice, then check an equation that links y, y′ and y″.
Rewrite first, simplify inverse trig, logarithmic and parametric differentiation.
Prices that jump at a slab, a speedometer reading, a balloon filling with air, a ladder sliding down a wall, bacteria multiplying and a ball in flight: each is one idea from this chapter.
A car park charges ₹40 for each hour or part of an hour. The fee is 40 × (the number of started hours), a staircase like [x] (turned the other way). Time flows smoothly, but the fee does not: one second before 2 hours you pay ₹80, one second after you pay ₹120.
The distance travelled s(t) grows as time passes. The speed at an instant is the slope of the distance-time graph at that instant: ds/dt. The rate at which the speed changes, the acceleration, is the second derivative d²s/dt².
A pump pushes in 400 cm³ of air every second. The volume is V = (4/3)πr³, so the chain rule gives dV/dt = 4πr² · dr/dt. The radius therefore grows at dr/dt = 400/(4πr²): as the balloon gets bigger, the same air spreads over a larger skin.
A 5 m ladder leans on a wall: x² + y² = 25, where x is the foot's distance from the wall and y the height of the top. Differentiate with respect to time: 2x dx/dt + 2y dy/dt = 0, so dy/dt = −(x/y) dx/dt. Pulling the foot at 0.5 m/s makes the top fall faster and faster as it nears the floor.
Each bacterium splits at a steady rate, so the more there are, the faster the colony grows: dN/dt = kN. The function with this property is N = N₀ekt, because (ekt)′ = k ekt. The slope of the graph is always k times its height.
A ball thrown at speed v and angle α has x = (v cos α)t and y = (v sin α)t − ½gt². Its direction of travel is dy/dx = (dy/dt)/(dx/dt) = tan α − gt/(v cos α). At the top of the flight the vertical speed is 0, so the path is momentarily flat.
Decide first, then tap each card.
All 137 questions of the chapter have an animated walkthrough (this chapter has no multiple-choice questions). Pick a question from the chips above each player. Press Play to watch it solved, or step through with Next. Each step has a short "why" under it and links to the basics it needs. Almost every derivative ends with a graph check: the tangent's slope is measured on the curve and compared with the answer.
Walk towards each point from the left and from the right and compare with the value. Where a constant is unknown, slide it until the red gap closes.
Split the function into machines, multiply their rates, put the inside back. Q9–Q10 test chord slopes from both sides of a corner or a jump.
Q1–Q8: differentiate every term, tag each y-term with dy/dx, collect, divide. Q9–Q15: a trig substitution turns the inside into one trig function; check that the angle lands in the principal range.
(eˣ)′ = eˣ and (log x)′ = 1/x, combined with the chain, product and quotient rules.
Logs turn products into sums and bring exponents down. For a sum of powers, take logs of each part separately.
dy/dx = (dy/dt) ÷ (dx/dt). Watch the point move along the curve with its horizontal and vertical speeds.
Differentiate twice. Q11–Q17 prove an equation linking y, y′ and y″; the green curve on the graph is that equation measured straight from the graph of y.
Rewrite first, then pick the method: chain, quotient, logs, implicit, parametric or twice.
Calculus rests on one new idea, the limit, plus old tools used in a new way: functions, slopes, trig, powers and logs, and a little algebra. Each step below starts from the question "why is it done this way?". Every example and exercise links here exactly where a basic is needed, and a button takes you straight back to the problem.
Some questions cannot be answered by just "plugging in". f(x) = (x² − 1)/(x − 1) gives 0/0 at x = 1, which is meaningless. Yet at x = 0.9, 0.99, 0.999 the values are 1.9, 1.99, 1.999, and at 1.1, 1.01, 1.001 they are 2.1, 2.01, 2.001. They are heading for 2.
We write limx→c f(x) = L when f(x) can be made as close to L as we like by taking x close enough to c (but not equal to c). The value at c itself plays no part: it may be missing, or be something else entirely.
A slope needs two points. At a single point, "rise over run" is 0/0. Limits let us ask what slope the two-point slopes are heading for as the points merge (step 5). And a graph is unbroken exactly when its height near c heads for the height at c (continuity).
A limit is a prediction: "if you keep walking towards c, this is the height you will find". Continuity asks: does the prediction come true when you arrive?
On the number line there are two ways to get close to c: from the left (x < c, written x → c⁻) and from the right (x > c, x → c⁺). The left-hand limit (LHL) and right-hand limit (RHL) are the heights reached each way. The ordinary limit exists only when they are equal.
To find a one-sided limit of a piecewise function, use the rule that applies on that side of c, and substitute c (if that rule is a polynomial or another continuous function).
A function is a rule giving one output for each input in its domain (see Chapter 1). Sometimes the rule changes from one part of the domain to another: f(x) = x + 2 for x ≤ 1 and x − 2 for x > 1. Read it like a tax table: first find which range x is in, then use that rule.
|x| = x when x ≥ 0 and −x when x < 0. It makes negatives positive, so |−3| = 3. Its graph is a V with a corner at 0.
[x] rounds down to an integer: [2.7] = 2, [5] = 5, [−0.4] = −1 (not 0, because −1 is the largest integer that is still ≤ −0.4). Its graph is a staircase.
If the rules skip a point (say they cover x < 0 and x > 0 only), that point is not in the domain. It is not a break; the question of continuity simply does not arise there.
sin(x²) means: square x, then take the sine. We write it f(g(x)) or (f ∘ g)(x), with g = square (inside) and f = sin (outside). Reading from the inside out is the key skill for the chain rule: find the innermost expression, then work outwards.
Continuity passes along the chain: if each machine turns small input changes into small output changes, so does the whole row. Rates multiply along the chain: if the inner machine doubles small changes and the outer triples them, the whole row multiplies them by 6. That is the chain rule.
The slope of a straight line is rise ÷ run: how much y changes for each unit of x. A curve has no single slope, so zoom in.
The chord from (c, f(c)) to (c + h, f(c + h)) has slope [f(c + h) − f(c)]/h. As h shrinks, the chord turns into the tangent, and its slope is the derivative:
dy/dx, f′(x) and y′ all mean the same thing: the rate of change of y per unit change of x. The second derivative y″ is the rate of change of the slope itself: positive when the slope is increasing (the graph bends up).
You rarely compute limits by hand, because the rules below package them once and for all.
(x + h)ⁿ = xⁿ + nxⁿ⁻¹h + (terms with h², h³…). Subtract xⁿ, divide by h, let h → 0: only nxⁿ⁻¹ survives.
Picture a rectangle u × v. When x grows a little, the area gains a strip v·Δu and a strip u·Δv (plus a tiny corner Δu·Δv that vanishes).
On the unit circle, a tiny turn Δx moves the point along an arc of length Δx; its height grows by about cos x · Δx.
The other rules you will use, all found the same way:
Combining
Trigonometric
A constant has derivative 0, and constant factors come out in front.
On the unit circle a point at angle θ is (cos θ, sin θ). Its distance from the centre is 1, so sin²θ + cos²θ = 1 (Pythagoras). Everything else follows from this and the addition formulas.
Addition
Double angle, in terms of tan θ
Half angle: 1 ± cos θ
sin⁻¹ returns angles in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2).
So sin⁻¹(sin θ) = θ only if θ is already in [−π/2, π/2]; otherwise you get a different angle with the same sine. Always check this after a substitution such as x = tan θ.
bⁿ means n copies of b multiplied together. The logarithm undoes it: logb a = x means bˣ = a (log₂ 8 = 3 because 2³ = 8).
Adding exponents multiplies powers. That single rule forces the other three below, and extends bˣ to every real x.
Exponent rules
Because powers multiply by adding exponents, logs turn products into sums:
Log laws
Only positive numbers have logs, since every power of a positive base is positive.
Put money in a bank at 100% a year, credited in n parts: you end with (1 + 1/n)ⁿ times your money. As n grows this settles on e = 2.71828… (also 1 + 1/1! + 1/2! + …).
It is the base whose curve has slope equal to its height, so (eˣ)′ = eˣ. Its inverse, the natural log, has (log x)′ = 1/x. In this chapter "log" means base e.
After implicit differentiation you get an equation where dy/dx appears in several terms, such as 2x + x·□ + y + 2y·□ = 0 with □ = dy/dx. Solve it like any linear equation: move every □-term to one side and everything else to the other (changing sign when crossing =), take □ out as a common factor, and divide.
Other algebra you will meet, plus a common denominator to add fractions:
The first helps you cancel a factor. Writing roots as powers lets the power rule handle them.
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