Understanding Differentials

Slopes · Rates of Change · Limits · Derivatives · Applications

Starting Point

The Language of Change

Differential calculus is the systematic study of how quantities change. The central object — the derivative — is simultaneously a slope, a rate, and a limit. This guide builds each idea from the ground up, while also giving you the vocabulary you'd otherwise hunt through multiple textbooks to find.

Every section is colour-coded so you can track the conceptual thread at a glance.

Colour Guide

Slopes & Linear geometry Rates of Change Limits & Continuity Derivatives & Rules Applications Practice
The Fundamental Definition  —  \[ f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} \]

① Slopes

  • Rise over run
  • Slope–intercept form
  • Point-slope form
  • Secant lines

② Rates of Change

  • Average rate
  • Difference quotient
  • Secant → tangent

③ Limits

  • ε–δ definition
  • One-sided limits
  • L'Hôpital's rule
  • Continuity

④ Derivatives

  • Power / product / quotient rules
  • Chain rule
  • Implicit diff.
  • Higher-order

⑤ Applications

  • Optimisation
  • Related rates
  • Kinematics
  • Linear approximation

⑥ Jargon Glossary

  • 50+ terms defined
  • Etymology notes
  • Cross-references
Section One

Slopes & Linear Functions

Definition — Slope The slope (symbol m) of a line measures steepness as the ratio of vertical change (rise) to horizontal change (run) between any two distinct points.
\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]
Rise: 0  |  Run: 2  |  m = 0  |  Eq: y = 0
1.0
Forms of a Linear Equation
NameFormWhen to use
Slope–intercept\(y = mx + b\)Know slope & y-intercept
Point–slope\(y - y_1 = m(x - x_1)\)Know slope & one point
Standard\(Ax + By = C\)Integer coefficients
Two-point\(\dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}\)Know two points
⚠ Common Pitfall — Undefined Slope Vertical lines (x = c) have undefined slope, not zero slope. Zero slope belongs to horizontal lines (y = c).
Step-by-Step — Slope from Two Points
  1. Label your points: \((x_1, y_1)\) and \((x_2, y_2)\).
  2. Subtract y-values: \(\Delta y = y_2 - y_1\) (rise).
  3. Subtract x-values: \(\Delta x = x_2 - x_1\) (run). Order must match.
  4. Divide: \(m = \Delta y / \Delta x\).
  5. Plug into point–slope form to find the full equation.
Real World — Grade of a Road A road that rises 8 m over 100 m of horizontal distance has a grade of \(8/100 = 0.08\), written as 8%. Highway engineers use this exact formula.
Section Two

Rates of Change

Definition — Average Rate of Change For a function \(f\) over an interval \([a,b]\), the average rate of change (ARC) is the slope of the secant line connecting \((a, f(a))\) and \((b, f(b))\).
\[ \text{ARC} = \frac{f(b) - f(a)}{b - a} = \frac{\Delta f}{\Delta x} \]
A = (-2, 0)  |  B = (2, 0)  |  ARC = 0
-2.0
2.0
Definition — Difference Quotient The difference quotient expresses the ARC as \(x + h\): \[ \frac{f(x+h) - f(x)}{h} \] Taking \(h \to 0\) yields the derivative.
⚠ Secant vs Tangent A secant line crosses a curve at two points (slope = ARC). A tangent line touches at exactly one point (slope = derivative).
Real World — Average vs Instantaneous Velocity A car travels 120 km in 1.5 hours → average speed 80 km/h (ARC of position). The speedometer reads instantaneous speed (derivative of position).
Key Theorem — Mean Value Theorem (MVT) If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists at least one \(c \in (a,b)\) such that: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
Section Three

Limits & Continuity

Definition — Limit (Informal) \(\lim_{x \to c} f(x) = L\) means: as \(x\) gets arbitrarily close to \(c\) (but never equals \(c\)), \(f(x)\) gets arbitrarily close to \(L\).
Definition — Limit (ε–δ Formal) For every \(\varepsilon > 0\) there exists \(\delta > 0\) such that \( 0 < |x - c| < \delta \implies |f(x) - L| < \varepsilon \).
Approaching x = 0.0  |  f(x) → 0.0  |  Limit = 0.0
0.0
⚠ Indeterminate Forms When direct substitution gives one of these, the limit is not immediately determined: \[ \frac{0}{0}, \quad \frac{\infty}{\infty}, \quad 0 \cdot \infty, \quad \infty - \infty, \quad 0^0, \quad 1^\infty, \quad \infty^0 \] Techniques: factoring, rationalising, L'Hôpital's Rule, Taylor series.
L'Hôpital's Rule If \(\lim f(x) = 0\) and \(\lim g(x) = 0\) (or both \(\pm\infty\)), and \(g'(x) \neq 0\) near \(c\): \[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
Continuity — Three Conditions \(f\) is continuous at \(c\) if and only if all three hold:
  1. \(f(c)\) is defined
  2. \(\lim_{x \to c} f(x)\) exists
  3. \(\lim_{x \to c} f(x) = f(c)\)
Standard Limits Worth Memorising
LimitValueWhy it matters
\(\lim_{x \to 0} \dfrac{\sin x}{x}\)1Foundation of trig derivatives
\(\lim_{x \to 0} \dfrac{1-\cos x}{x}\)0Cos derivative derivation
\(\lim_{x \to 0} (1+x)^{1/x}\)\(e\)Definition of Euler's number
\(\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x\)\(e\)Alternative e definition
\(\lim_{x \to 0} \dfrac{e^x - 1}{x}\)1Exponential derivative
Section Four

Derivatives

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Notation Rosetta Stone All of these mean "derivative of \(y = f(x)\) with respect to \(x\)":
NotationAuthorBest used for
\(f'(x)\)Lagrange (1797)General functions
\(\dfrac{dy}{dx}\)Leibniz (1684)Related rates
\(\dot{y}\)Newton (1671)Time derivatives in physics
\(D_x f\)Euler / CauchyOperator notation; ODEs
\(\partial f / \partial x\)—Partial derivatives
x = 0  |  f(x) = 0  |  f′(x) = 0
Tangent: y = 0
0.0
f(x)   Tangent   f′(x)

Differentiation Rules

Power Rule \[ \frac{d}{dx}[x^n] = nx^{n-1} \] Works for any real \(n\). Most frequently used rule.
Product Rule \[ (uv)' = u'v + uv' \] "first · d-second + second · d-first."
Quotient Rule \[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \] "low d-high minus high d-low over low²."
Chain Rule \[ [f(g(x))]' = f'(g(x)) \cdot g'(x) \] "Derivative of outside × inside."
Trig Derivatives \( (\sin x)' = \cos x \)
\( (\cos x)' = -\sin x \)
\( (\tan x)' = \sec^2 x \)
\( (\sec x)' = \sec x \tan x \)
Exp & Log \( (e^x)' = e^x \)
\( (a^x)' = a^x \ln a \)
\( (\ln x)' = 1/x \)
\( (\log_a x)' = \tfrac{1}{x \ln a} \)
Higher-Order Derivatives The second derivative \(f''(x)\) measures concavity and acceleration.
OrderNotationPhysics meaning
1st\(s'(t)\)Velocity
2nd\(s''(t)\)Acceleration
3rd\(s'''(t)\)Jerk
4th\(s^{(4)}(t)\)Snap / Jounce
⚠ Differentiability vs Continuity Differentiability implies continuity, but not vice versa. Example: \(f(x) = |x|\) is continuous everywhere but not differentiable at \(x = 0\).
Implicit Differentiation When \(y\) cannot be isolated (e.g., \(x^2 + y^2 = 25\)):
  1. Differentiate both sides w.r.t. \(x\).
  2. Treat \(y\) as a function of \(x\): \(\tfrac{d}{dx}[y^2] = 2y\tfrac{dy}{dx}\).
  3. Collect all \(\tfrac{dy}{dx}\) terms and solve.
Result: \(\tfrac{dy}{dx} = -\tfrac{x}{y}\) for the circle above.
Section Five

Applications of Derivatives

Critical Points, Extrema, Inflection Points
TermConditionMeaning
Critical point\(f'(c) = 0\) or DNECandidate for local max/min
Local minimum\(f'(c)=0\), \(f''(c)>0\)Bowl-shaped
Local maximum\(f'(c)=0\), \(f''(c)<0\)Hill-shaped
Inflection point\(f''(c)=0\), sign changeConcavity changes
Saddle point\(f'(c)=0\), \(f''(c)=0\)Neither max nor min
First & Second Derivative Tests 1st Derivative Test: If \(f'\) changes from + to − at \(c\) → local max; − to + → local min.
2nd Derivative Test: At \(f'(c)=0\): \(f''(c) > 0\) → local min; \(f''(c) < 0\) → local max.

Optimisation Problems

General Strategy
  1. Identify the quantity to maximise/minimise. Write objective function \(Q\).
  2. Use constraints to eliminate variables until \(Q\) is in one variable.
  3. Find \(Q'\), set to zero, solve critical points.
  4. Verify using 2nd derivative test or boundary values.
  5. State the answer with units.

Related Rates

Concept When two quantities are related by an equation, differentiating both sides w.r.t. time \(t\) links their rates. This is the chain rule in action.

Linear Approximation & Differentials

\[ f(x) \approx f(a) + f'(a)(x - a) \quad \text{for } x \approx a \]

The tangent-line approximation: near any differentiable point, a curve looks like its tangent line.

Definition — Differential The differential of \(y = f(x)\) is \(dy = f'(x)\,dx\). This formalises Leibniz notation and underpins numerical methods and error analysis.
Real World — Error Propagation Sphere radius \(r = 5\) cm, error \(\pm 0.02\) cm. Volume error: \(dV = 4\pi r^2 \, dr = 4\pi(25)(0.02) \approx 6.28 \text{ cm}^3\).

Kinematics — Position, Velocity, Acceleration

QuantitySymbolRelationSign
Position\(s(t)\)GivenLocation
Velocity\(v(t) = s'(t)\)1st deriv.+ = right
Speed\(|v(t)|\)Magnitude≥ 0
Acceleration\(a(t) = s''(t)\)2nd deriv.+ = speeding up
Section Six

Practice Problems

Problem 1 of 3 — Slopes

Challenge Problems

Stretch — Try Without Hints
  1. Find the derivative of \(f(x) = x^x\). (Hint: use logarithmic differentiation.)
  2. Find all points where the tangent to \(y = x^3 - 3x\) is horizontal.
  3. Evaluate \(\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{\sin(5x)}\).
  4. A 10 m ladder leans against a wall. Base slides at 0.5 m/s. How fast does the top slide when base is 6 m out?
  5. Show \(f(x) = |x|\) is continuous but not differentiable at \(x = 0\).