§ 01What Is Differentiation?

Differentiation is the process of finding the instantaneous rate of change of a function: the slope of its graph at any given point. It's arguably the single most useful tool in applied mathematics.

Imagine you are watching a car move along a road. You can measure its average speed over any time interval by dividing distance by time. But what is its speed at exactly 2:00:00 pm, not over a second, not over a millisecond, but at that precise instant? That is a derivative.

The derivative of a function f at a point x is defined as the limit of the average rate of change as the interval shrinks to zero:

The Derivative — Limit Definition

f′(x) = lim_h→0 [f(x+h) − f(x)] / h

This is the slope of the tangent line at x. It exists whenever the function is smooth (differentiable) at that point.

In practice, we almost never use this limit directly: the rules in this section let us differentiate any function built from polynomials, trig, exponentials, and logs in seconds. But the limit is the foundation everything rests on, and understanding it makes every rule feel inevitable rather than arbitrary.

Differentiation and Integration Are Two Sides of the Same Coin The Fundamental Theorem of Calculus, one of the great results in mathematics, says that differentiation and integration are inverse operations. Every time you differentiate in this section, you're also building the intuition you'll need to reverse the process later on. Get comfortable here and integration will feel far more natural when you get to it.

§ 02All 13 Differentiation Topics

Each page below contains full explanations, multiple worked examples, and a 10-question interactive quiz. Start at the top and work down, or jump to whichever topic you need.

01 · Foundation Differentiation from First Principles Use the limit definition directly. Derive d/dx(x²), d/dx(sin x), and more from scratch — no rules, just the definition. 12 examples · quiz 02 · Core Rules Differentiation Rules Constant rule, power rule, sum/difference rule, constant multiple rule. The toolkit every other technique builds on. 12 examples · quiz 03 · Products The Product Rule (uv)′ = u′v + uv′. Why you cannot just multiply derivatives, with proof from first principles and 12 worked examples. 12 examples · quiz 04 · Quotients The Quotient Rule (u/v)′ = (u′v − uv′)/v². The "lo-d-hi minus hi-d-lo" rule with proof and comparison to product rule alternatives. 12 examples · quiz 05 · Composition The Chain Rule dy/dx = (dy/du)⋅(du/dx). The most-used rule in calculus. Single, double and triple chains with 15 worked examples. 15 examples · quiz 06 · Implicit Implicit Differentiation Differentiate when y cannot be isolated. Circles, ellipses, and curves defined implicitly — including tangent lines and second derivatives. 12 examples · quiz 07 · Higher Order Higher-Order Derivatives f′′, f′′′, f⁽ⁿ⁾ — all notation systems, physical meaning (velocity → acceleration → jerk), concavity, and n^th-derivative patterns. 10 examples · quiz 08 · Trigonometry Trig Derivatives d/dx(sin x), cos x, tan x, sec x, csc x, cot x — all six, derived and applied with the chain rule and worked examples. 12 examples · quiz 09 · Inverse Trig Inverse Trig Derivatives d/dx(arcsin x), arccos x, arctan x and their compound forms. Proofs using implicit differentiation and Pythagorean identities. 12 examples · quiz 10 · Logarithms Log Derivatives d/dx(ln x) = 1/x and d/dx(log_a x) = 1/(x ln a). Chain rule applications, absolute value in ln|x|, and logarithmic properties that simplify work. 12 examples · quiz 11 · Exponentials Exponential Derivatives d/dx(e^x) = e^x — why it’s magical. d/dx(a^x) = a^x⋅ln a. Chain rule with compound exponents, 12 examples including product and quotient combinations. 12 examples · quiz 12 · Parametric Parametric Differentiation When x = f(t) and y = g(t), find dy/dx = (dy/dt)/(dx/dt). Second derivatives, tangents and normals on parametric curves. 10 examples · quiz 13 · Advanced Logarithmic Differentiation Differentiate x^x, (sin x)^{cos x}, and other functions where both base and exponent vary. Take ln of both sides, differentiate, multiply back. 10 examples · quiz

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§ 03Suggested Learning Path

Differentiation topics build on each other in a logical sequence. The path below groups them by what prerequisite knowledge each page requires. If you’re studying from scratch, follow stage by stage. If you’re reviewing, jump to the stage that covers your gap.

Stage 1

Prerequisites: Limits & algebra First Principles Differentiation Rules

Stage 2

Prerequisites: Stage 1 Product Rule Quotient Rule Chain Rule

Stage 3

Prerequisites: Stage 2 · Trig functions Trig Derivatives Log Derivatives Exponential Derivatives

Stage 4

Prerequisites: Stages 2 & 3 Implicit Differentiation Higher-Order Derivatives Inverse Trig Derivatives

Stage 5

Prerequisites: All prior stages Parametric Differentiation Logarithmic Differentiation

§ 04Quick Reference — Every Rule at a Glance

Bookmark this section. Once you have worked through all the topic pages, these cards form your complete differentiation cheat-sheet. Every formula here is proved in full on its dedicated page.

Constant Rule

d/dx (c) = 0 Any constant differentiates to zero — its graph is flat, slope zero everywhere.

Power Rule

d/dx (xⁿ) = nxⁿ⁻¹ Works for any real n: integer, fraction, negative, zero.

Constant Multiple

d/dx [c⋅f(x)] = c⋅f′(x) Constants factor out of derivatives.

Sum / Difference

d/dx [f ± g] = f′ ± g′ Differentiate term by term.

Product Rule

(uv)′ = u′v + uv′ “First⋅d(Second) + Second⋅d(First).” Cannot simply multiply derivatives.

Quotient Rule

(u/v)′ = (u′v − uv′) / v² lo⋅d(hi) − hi⋅d(lo), all over lo-squared.

Chain Rule

d/dx[f(g(x))] = f′(g(x))⋅g′(x) Outer derivative (leave inner alone) times derivative of inner.

Natural Exponential

d/dx (e^x) = e^x The only function equal to its own derivative.

General Exponential

d/dx (a^x) = a^x⋅ln a The ln(a) factor is the “tax” for using a base other than e.

Natural Logarithm

d/dx (ln x) = 1/x Valid for x > 0. Use ln|x| for x ≠ 0, giving d/dx(ln|x|) = 1/x.

Log Base a

d/dx (log_a x) = 1 / (x ln a) Derived by converting to natural log: log_a x = ln x / ln a.

Sine / Cosine

d/dx (sin x) = cos x d/dx (cos x) = −sin x Note the minus sign on cos. Easy to forget under pressure.

Tangent / Secant

d/dx (tan x) = sec² x d/dx (sec x) = sec x tan x Both derived using quotient rule from sin/cos.

Cosecant / Cotangent

d/dx (csc x) = −csc x cot x d/dx (cot x) = −csc² x Mirror images of sec and tan, both with negative signs.

arcsin / arccos

d/dx (arcsin x) = 1/√(1−x²) d/dx (arccos x) = −1/√(1−x²) Proved by implicit differentiation on sin(y)=x.

arctan / arccot

d/dx (arctan x) = 1/(1+x²) d/dx (arccot x) = −1/(1+x²) No square root in the denominator — cleaner than arcsin/arccos.

§ 05Where Differentiation Leads

Mastering differentiation unlocks the next tier of calculus. The derivative is not just a calculation technique, it is the core of optimisation, curve analysis, differential equations, and multivariable calculus. Once you are fluent here, these sections will feel like natural extensions.

Apply It

Applications of Differentiation

Reverse It

Integration

Model With It

Differential Equations

From This Section to Applications — a Direct Line Once you can reliably find f′(x) for any function, the next section — Applications of Differentiation — shows you how to use that derivative to find maxima and minima, sketch curves, solve optimisation problems, and prove classical results like Rolle’s Theorem and the Mean Value Theorem. The derivative is the engine; applications show you what to build with it.