Chapter 3 established what a derivative is: the instantaneous rate of change of a function at a point, defined as a limit of difference quotients. That definition is the foundation of everything in differential calculus. It is rigorous, conceptually clear, and exact. But applying the definition from scratch every time a derivative is needed is slow, increasingly laborious as functions grow complex, and ultimately impractical in applied work.

Mathematicians recognized this limitation early and derived a collection of differentiation rules — systematic shortcuts that are themselves proven from the limit definition — that allow you to differentiate most functions quickly and reliably. These rules are not arbitrary tricks. Each one encodes a precise limit calculation that has already been carried out in full generality.

This chapter teaches those rules. You will learn how to differentiate constants, power functions, polynomials, products, quotients, and compositions of functions. By the end, you should be able to differentiate a wide class of algebraic functions, identify which rule applies to a given function before computing, and interpret every derivative you find as a meaningful rate of change.

The goal is fluency without mindlessness. A student who can execute differentiation rules but has no idea what the result means has not finished the problem. Every derivative this chapter asks you to find is a rate of change. Its sign tells you whether something is increasing or decreasing. Its magnitude tells you how sensitive the output is to changes in the input. Its units connect the abstract symbol to a real measurement. These rules are tools; context gives them meaning.

Differentiation rules are the working vocabulary of applied calculus. Physics uses derivatives of position to find velocity and acceleration. Chemistry uses them to study reaction rates. Economics uses them for marginal cost, marginal revenue, and marginal profit. Biology uses them for growth and population dynamics. Engineering uses them to analyze stress, heat flow, and signal behavior. Machine learning uses them in gradient descent — the algorithm that trains neural networks by minimizing a loss function.

None of these applications can pause for a from-scratch limit calculation every time a derivative is needed. Differentiation rules make calculus operational. This chapter gives you those rules and, more importantly, develops the judgment to use them correctly.

In Chapter 3, you learned that the derivative of \(f\) at \(x\) is defined as

This definition is the rigorous foundation of differential calculus. It is exact, and it is the reason every rule in this chapter is trustworthy. But applying the limit definition from scratch every time is slow and error-prone. To differentiate \(f(x) = x^5\) from the definition alone, you would need to expand \((x+h)^5\) using the binomial theorem, subtract \(x^5\), divide by \(h\), simplify, and then evaluate the limit. That calculation is manageable for \(x^5\) but becomes overwhelming for \(x^{20}\), or for a product like \((3x^2 + 1)(x^4 - 5x)\), or for a composition like \((x^3 + 2)^{10}\).

Differentiation rules solve this problem. Each rule was derived from the limit definition — but once proven to hold in full generality, it can be applied directly to any function matching its form. Using the Power Rule, the derivative of \(x^{20}\) is \(20x^{19}\) — one step, no expansion needed.

This efficiency is not a shortcut that sacrifices understanding. The rules encode the underlying limit behavior; they simply compress the work of calculations that are guaranteed to give the same result every time. The real skill this chapter develops is rule identification: looking at a function before computing and asking, "What is the structure here?" Is it a constant? A power? A product? A quotient? A composition? The structure determines the rule. Identifying structure before computing saves errors and builds mathematical judgment.

The simplest differentiation rule concerns constant functions — functions of the form \(f(x) = c\), where \(c\) is any fixed number. A constant function is a horizontal line. Its output does not change as \(x\) changes. Therefore, its rate of change is zero everywhere. This geometric fact is what the Constant Rule formalizes.

Derivation: If \(f(x) = c\), then \(f(x+h) = c\) as well, since a constant does not depend on \(x\). Therefore:

The Power Rule is the most-used differentiation rule in first-semester calculus. It applies to any function of the form \(f(x) = x^n\), where \(n\) is a real number.

Why it works (integer case): For a positive integer \(n\), expanding \((x+h)^n\) with the binomial theorem yields \(x^n + nx^{n-1}h + \text{(terms with } h^2 \text{ and higher)}\). Subtracting \(x^n\), dividing by \(h\), and taking the limit as \(h \to 0\) leaves only \(nx^{n-1}\). The same result extends to all real-number exponents through related arguments.

The Power Rule applies to positive integer powers (\(x^2, x^3, x^{10}\)), negative integer powers (\(x^{-1}, x^{-3}\)), rational exponents (\(x^{1/2}, x^{2/3}\)), and zero (\(\frac{d}{dx}[x^0] = \frac{d}{dx}[1] = 0\), consistent with the Constant Rule). It does not apply to exponential functions like \(2^x\) or \(e^x\), where the variable is in the exponent rather than the base.

When a function is multiplied by a constant, the derivative is also multiplied by that constant. The constant factor simply passes through differentiation unchanged.

Derivation: From the limit definition, \(\dfrac{d}{dx}[c \cdot f(x)] = \displaystyle\lim_{h \to 0} \dfrac{c \cdot f(x+h) - c \cdot f(x)}{h} = c \cdot \displaystyle\lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} = c \cdot f'(x)\). The constant factor passes through the limit operator because limits respect scalar multiplication.

Functions often consist of several terms added or subtracted. The Sum and Difference Rules allow you to differentiate each term independently and add or subtract the results.

A polynomial function is a sum of terms of the form \(a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\), where all exponents are non-negative integers. Combining the Constant Rule, Power Rule, Constant Multiple Rule, and Sum/Difference Rules, you can differentiate any polynomial term by term. There is no special rule for polynomials — you already have all the tools.

One important pattern: the derivative of a polynomial of degree \(n\) is a polynomial of degree \(n - 1\). Each differentiation reduces the degree by one. Constant terms vanish; linear terms become constants; quadratics become linear; and so on.

Suppose \(h(x) = f(x) \cdot g(x)\) — the product of two functions. The natural question is: can you differentiate by simply multiplying the individual derivatives? The answer is no. This is one of the most important facts in differential calculus.

Consider \(f(x) = x^2\) and \(g(x) = x^3\). Their product is \(h(x) = x^5\), so \(h'(x) = 5x^4\). But the product of the derivatives is \(f'(x) \cdot g'(x) = 2x \cdot 3x^2 = 6x^3 \neq 5x^4\). Multiplying derivatives gives the wrong answer. The correct rule is more nuanced.

Suppose \(h(x) = \dfrac{f(x)}{g(x)}\). As with products, you cannot simply divide the derivatives. The Quotient Rule gives the correct formula.

The Chain Rule is arguably the most important differentiation rule you will learn. It handles composite functions — functions of the form \(h(x) = f(g(x))\), where one function is applied to the output of another. Without the Chain Rule, differentiation is restricted to functions you can write as explicit sums, products, or quotients. With it, you can differentiate functions of enormous complexity built by nesting one function inside another.

Consider \(h(x) = (x^2 + 3)^4\). This is a power function applied to an inner expression \(x^2 + 3\). The Power Rule alone gives \(4(x^2+3)^3\) — but that is incomplete. Differentiating a composite function means accounting for how fast both the outer and the inner function are changing. The Chain Rule captures both.

Step 1: Identify the outer and inner functions.
Step 2: Differentiate the outer function with respect to the inner, leaving the inner function unchanged.
Step 3: Multiply the result by the derivative of the inner function.

Real functions often require more than one rule. A product of two composite functions requires both the Product Rule and the Chain Rule. A quotient whose numerator is a composition requires both the Quotient Rule and the Chain Rule. The key skill is reading the outermost structure of a function first, then working inward.

Strategy: Identify the outermost operation — sum, product, quotient, or composition. Apply the rule for that structure first. As you differentiate each piece, apply further rules as needed inside. When finished, simplify and factor the result, especially if you will later need to solve \(h'(x) = 0\).

You can differentiate a function more than once. The derivative of the derivative is the second derivative. Differentiating again yields the third derivative, and so on. These higher-order derivatives carry meaningful information about how a function changes.

The first derivative measures the rate of change of \(f\). The second derivative measures the rate of change of the first derivative — how quickly the rate of change itself is changing. In motion: if \(s(t)\) is position, then \(s'(t) = v(t)\) is velocity, and \(s''(t) = a(t)\) is acceleration. The second derivative also governs the concavity of a graph, a topic developed fully in Chapter 7.

Calculus uses several notation systems for derivatives. You will encounter all of them in this book, in other courses, and in applications. Understanding what each notation means — and when each is preferred — is part of mathematical fluency.

Function notation \(f'(x)\) emphasizes that the derivative is itself a function of \(x\). It is compact and common in pure mathematics.

Leibniz notation \(\dfrac{dy}{dx}\) makes the variables explicit. It is widely used in physics, engineering, and applied mathematics because it shows both what is being differentiated and with respect to which variable. The operator \(\dfrac{d}{dx}\) means "differentiate with respect to \(x\)."

Evaluating at a point: \(f'(a)\) or \(\left.\dfrac{dy}{dx}\right|_{x=a}\) both mean the value of the derivative when \(x = a\).

Non-standard variables: If time is the independent variable and position is \(s\), write \(\dfrac{ds}{dt}\) for velocity. If cost is \(C\) and quantity is \(q\), write \(\dfrac{dC}{dq}\) for marginal cost. Leibniz notation makes the variable relationship transparent in applied contexts.

The rules in this chapter are systematic, but students make a predictable set of errors when applying them. The mistakes below are not random — each reflects a specific misunderstanding. Studying them carefully is itself part of learning calculus.

This chapter covered differentiation rules for algebraic functions: functions built from powers, products, quotients, and compositions of polynomial and rational expressions. But the most important functions in science, engineering, finance, and data science extend far beyond algebra.

Exponential functions model growth and decay: compound interest, radioactive decay, population, bacteria. Logarithmic functions model scale, information theory, and entropy. Trigonometric functions model oscillation, waves, rotation, and alternating current. These are the transcendental functions you studied in Trigonometry and Precalculus, and calculus applies fully to all of them.

Chapter 5 develops the derivatives of these function families. The Chain Rule you learned in Section 4.9 will be essential throughout. Differentiating \(e^{3x^2}\) requires recognizing a composition and multiplying by the inner derivative. Differentiating \(\sin(x^2)\) requires the same pattern. Every Chain Rule problem you have practiced in this chapter is preparation for Chapter 5.

Before moving forward, make sure you are comfortable with: