Derivatives of cos x, sin x, eˣ, and ln x

SECTION 1

Historical Context & Motivation

The story of differentiating transcendental functions — sines, cosines, exponentials, and logarithms — is inseparable from the story of calculus itself. When Isaac Newton and Gottfried Wilhelm Leibniz independently developed the foundations of calculus in the late seventeenth century, polynomial functions were the first to yield their derivatives. However, the physical world demanded more: planetary orbits required trigonometric functions, compound growth required exponentials, and bridging the two demanded logarithms. The quest to differentiate these four function families drove some of the most elegant mathematics of the Enlightenment era and produced results that remain indispensable on the AP Calculus AB exam today.

1665–1676

Newton & Leibniz Develop Calculus

Newton uses 'fluxions' to study motion while Leibniz introduces the dx/dy notation. Both work primarily with polynomials and power series, setting the stage for extending differentiation to transcendental functions.

1748

Euler's Introductio in Analysin Infinitorum

Leonhard Euler formalizes the exponential function eˣ and its inverse ln x, showing that e ≈ 2.71828 is the unique base for which the derivative of the exponential equals itself — a property that became central to analysis.

1750s

Trigonometric Derivatives Established

Building on Euler's work connecting exponentials to trigonometry via eⁱˣ = cos x + i sin x, mathematicians rigorously prove that d/dx(sin x) = cos x and d/dx(cos x) = −sin x using limit arguments involving the squeeze theorem.

1821

Cauchy's Rigorous Limits

Augustin-Louis Cauchy publishes Cours d'Analyse, placing derivatives on the firm footing of the ε–δ definition of limits. This finally provides airtight proofs for the derivative formulas students learn today.

The central question these mathematicians answered was deceptively simple: if a quantity oscillates sinusoidally or grows exponentially, how fast is it changing at any given instant? The four derivative formulas you will master in this lesson — for sin x, cos x, eˣ, and ln x — are the answers to that question, and they underpin virtually every applied calculus problem from modeling population growth to analyzing alternating current circuits.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it is important to understand why these four functions are grouped together. They are all transcendental functions — functions that cannot be expressed as finite combinations of algebraic operations (addition, multiplication, roots). Unlike polynomials, whose derivatives always reduce the degree by one, transcendental derivatives cycle or reproduce themselves, which is precisely what makes them so powerful in modeling real-world phenomena.

d/dx (sin x) = cos x

The rate of change of the sine function at any point equals the cosine of that point. The derivative is phase-shifted by π/2 radians to the left.

d/dx (cos x) = −sin x

The cosine derivative introduces a negation. This sign change reflects the fact that cos x is decreasing when sin x is positive and vice versa.

d/dx (eˣ) = eˣ

The exponential function with base e is its own derivative — the only nontrivial function family with this self-replicating property. This makes eˣ the natural choice for modeling uninhibited growth and decay.

d/dx (ln x) = 1/x

The natural logarithm's derivative is the reciprocal function, defined for x > 0. This result connects logarithmic and rational functions and appears in integration as the antiderivative of 1/x.

✦ KEY TAKEAWAY

Think of these four derivative rules as a toolkit of fundamental 'gears' in a machine. Just as an engineer selects specific gears to convert one type of motion into another, a calculus student selects these derivative formulas to convert a function into its rate of change. The trigonometric pair (sin and cos) act like interlocking gears that cycle into each other, while eˣ is the rare self-driving gear — it powers itself. Meanwhile, ln x translates exponential behavior into algebraic terms (1/x), bridging two different mathematical languages.

A critical prerequisite for the trigonometric derivatives is the radian measure requirement: the formulas d/dx(sin x) = cos x and d/dx(cos x) = −sin x hold only when x is measured in radians. If x were in degrees, an additional conversion factor of π/180 would appear, breaking the elegant simplicity. This is the fundamental reason calculus courses insist on radian mode — it is not merely a convention but a mathematical necessity rooted in the limit lim(h→0) (sin h)/h = 1, which is true exclusively in radians.

SECTION 3

Visual Explanation — Functions and Their Derivatives

The most illuminating way to grasp these derivative relationships is to see each function graphed alongside its derivative. In the diagram below, observe how the slope of the original function at every point matches the height of the derivative curve. Where sin x reaches its peak (slope = 0), cos x crosses zero; where eˣ has a certain height, its derivative has exactly the same height.

Three panels showing each function (solid) paired with its derivative (dashed or identical). Notice that sin x and cos x are phase-shifted copies of each other, eˣ overlaps perfectly with its own derivative, and ln x's derivative 1/x is a decreasing hyperbola that is steepest near x = 0⁺ where ln x rises most rapidly.

Pay particular attention to the sin x / cos x panel. At x = 0, sin x is rising through zero at its maximum rate; correspondingly, cos(0) = 1, confirming the derivative is at its maximum. At x = π/2, sin x reaches its peak and momentarily stops increasing — its slope is zero — and indeed cos(π/2) = 0. This visual correspondence between the height of the derivative and the slope of the original function is the single most important intuition to carry forward.

SECTION 4

Mathematical Framework — Proofs from the Limit Definition

Each of the four derivative formulas can be derived rigorously from the limit definition of the derivative: f′(x) = lim(h→0) [f(x + h) − f(x)] / h. Understanding at least the key steps of these proofs deepens your conceptual grasp and prepares you for the free-response questions on the AP exam that ask you to justify derivative rules from first principles.

Derivative of sin x

LIMIT DEFINITION APPLIED TO SIN X

d/dx (sin x) = lim(h→0) [sin(x + h) − sin x] / h

Expanding sin(x + h) = sin x cos h + cos x sin h via the angle-addition identity, then separating terms: = lim(h→0) [sin x (cos h − 1) + cos x sin h] / h = sin x · lim(h→0)(cos h − 1)/h + cos x · lim(h→0)(sin h)/h. Since lim(h→0)(sin h)/h = 1 and lim(h→0)(cos h − 1)/h = 0, we obtain d/dx(sin x) = cos x.

Derivative of cos x

LIMIT DEFINITION APPLIED TO COS X

d/dx (cos x) = lim(h→0) [cos(x + h) − cos x] / h

Using cos(x + h) = cos x cos h − sin x sin h: = lim(h→0) [cos x (cos h − 1) − sin x sin h] / h = cos x · 0 − sin x · 1 = −sin x. The negative sign is essential and a frequent source of errors on the AP exam.

Derivative of eˣ

LIMIT DEFINITION APPLIED TO eˣ

d/dx (eˣ) = lim(h→0) [e^(x+h) − eˣ] / h = eˣ · lim(h→0) (eʰ − 1) / h

Factor out eˣ since it does not depend on h. The remaining limit, lim(h→0)(eʰ − 1)/h, equals 1 — this is in fact one way to define the number e. Therefore d/dx(eˣ) = eˣ · 1 = eˣ.

Derivative of ln x

LIMIT DEFINITION APPLIED TO LN X

d/dx (ln x) = lim(h→0) [ln(x + h) − ln x] / h = lim(h→0) (1/h) · ln((x + h)/x)

Rewrite as lim(h→0) (1/h) · ln(1 + h/x). Let u = h/x so h = ux and as h→0, u→0: = lim(u→0) (1/(ux)) · ln(1 + u) = (1/x) · lim(u→0) ln(1 + u)/u. Since lim(u→0) ln(1 + u)/u = 1, the result is 1/x for x > 0.

⚠️ Two Critical Limits

The entire derivation of the trigonometric derivatives rests on two special limits: lim(h→0)(sin h)/h = 1 and lim(h→0)(cos h − 1)/h = 0. Both hold exclusively when h is in radians. The AP exam frequently tests whether students can identify and apply these limits, so commit them to memory alongside the derivative rules themselves.

SECTION 5

Detailed Breakdown — Patterns and Sign Behavior

A deeper look at these four derivatives reveals elegant patterns that simplify memorization and deepen understanding. The trigonometric derivatives cycle through a four-step pattern, the exponential is uniquely self-referential, and the logarithmic derivative bridges transcendental and algebraic worlds.

Left: The trigonometric derivative cycle — differentiating sin x yields cos x, then −sin x, then −cos x, and the fourth derivative returns to sin x. Right: eˣ maps to itself under differentiation (self-loop), while ln x maps to the algebraic function 1/x.

Successive derivatives of the four key functions

Function f(x)	f′(x)	f″(x)	f‴(x)	f⁽⁴⁾(x)
sin x	cos x	−sin x	−cos x	sin x
cos x	−sin x	−cos x	sin x	cos x
eˣ	eˣ	eˣ	eˣ	eˣ
ln x	1/x = x⁻¹	−x⁻²	2x⁻³	−6x⁻⁴

The table above reveals that the trigonometric derivatives repeat with period 4 — a fact frequently exploited in AP problems asking for the 50th or 101st derivative of sin x. To find the nth derivative of sin x, compute n mod 4: if the remainder is 0, the answer is sin x; if 1, cos x; if 2, −sin x; if 3, −cos x. The exponential function eˣ is invariant under all orders of differentiation, and the successive derivatives of ln x follow the power rule once you express 1/x as x⁻¹, generating factorials with alternating signs: (−1)ⁿ⁺¹(n − 1)! · x⁻ⁿ.

SECTION 6

Worked Example

Let us work through a multi-step problem that combines several of our derivative rules. This mirrors the level of complexity you will encounter in both the multiple-choice and free-response portions of the AP exam.

Differentiate f(x) = 3 sin x − 2eˣ + 5 ln x

Step 1 — Identify the StructureThe function f(x) = 3 sin x − 2eˣ + 5 ln x is a linear combination of three transcendental functions. By the constant multiple rule and the sum/difference rule, we can differentiate each term separately and preserve the coefficients.

Step 2 — Differentiate 3 sin xApplying d/dx(sin x) = cos x with the constant multiple rule: d/dx(3 sin x) = 3 cos x.

3 cos x

Step 3 — Differentiate −2eˣSince d/dx(eˣ) = eˣ: d/dx(−2eˣ) = −2eˣ.

−2eˣ

Step 4 — Differentiate 5 ln xSince d/dx(ln x) = 1/x: d/dx(5 ln x) = 5/x.

5/x

Step 5 — Combine ResultsAssembling all three terms: f′(x) = 3 cos x − 2eˣ + 5/x. This expression is valid for all x > 0 (the domain restriction comes from the ln x term, which requires x > 0).

f′(x) = 3 cos x − 2eˣ + 5/x

⚠️ Common Pitfall

Students frequently forget the negative sign when differentiating cos x, writing sin x instead of −sin x. Another frequent error is writing the derivative of ln x as 1/(ln x) instead of 1/x. Always double-check sign conventions and distinguish between the function and its derivative.

SECTION 7

Common Errors & Comparisons

Mastering these four derivatives means not only knowing the correct formulas but also understanding where students most commonly go wrong. The table below contrasts the correct derivative with the most frequent incorrect responses seen on the AP exam, along with the conceptual error behind each mistake.

Common derivative errors and their sources

Function	Correct Derivative	Common Error	Why It's Wrong
sin x	cos x	−cos x	Confusing the sign pattern; the negative appears when differentiating cos x, not sin x.
cos x	−sin x	sin x	Dropping the negative sign. Remember: cos x is decreasing when sin x is positive.
eˣ	eˣ	xeˣ⁻¹	Misapplying the power rule. The power rule applies to xⁿ, not to aˣ. The base is a constant and the exponent is the variable.
ln x	1/x	1/(ln x)	Confusing the function with its derivative. The ln disappears entirely upon differentiation.

✦ KEY TAKEAWAY

The single biggest source of confusion is mixing up the power rule (d/dx(xⁿ) = nxⁿ⁻¹) with the exponential derivative (d/dx(eˣ) = eˣ). Think of it this way: in xⁿ, the variable is the base and the exponent is fixed — you 'bring down' the exponent. In eˣ, the base is fixed and the exponent is the variable — the function regenerates itself. These are fundamentally different operations, like rotating a wheel (power rule reduces degree) versus a mirror reflecting itself (exponential reproduces).

SECTION 8

Connection to Advanced Theory

The four derivative rules you have learned form the foundation upon which more sophisticated techniques are built. In particular, the chain rule will extend each of these formulas to composite functions: d/dx(sin(g(x))) = cos(g(x)) · g′(x), d/dx(e^(g(x))) = e^(g(x)) · g′(x), and so on. The table below shows how each basic rule generalizes once the chain rule is introduced, a topic you will encounter very soon in your AP Calculus AB course.

Basic rules vs. chain-rule extensions

Basic Rule (this lesson)	With Chain Rule (upcoming)	Example
d/dx(sin x) = cos x	d/dx(sin u) = cos u · du/dx	d/dx(sin 3x) = cos 3x · 3 = 3 cos 3x
d/dx(cos x) = −sin x	d/dx(cos u) = −sin u · du/dx	d/dx(cos x²) = −sin x² · 2x
d/dx(eˣ) = eˣ	d/dx(eᵘ) = eᵘ · du/dx	d/dx(e⁻ˣ) = e⁻ˣ · (−1) = −e⁻ˣ
d/dx(ln x) = 1/x	d/dx(ln u) = (1/u) · du/dx	d/dx(ln(x² + 1)) = 2x/(x² + 1)

Beyond the chain rule, these derivatives connect to deep structural ideas in mathematics. Euler's formula, eⁱˣ = cos x + i sin x, reveals that the exponential and trigonometric functions are secretly unified through complex numbers. This unity explains why their derivatives share such elegant properties — they are all manifestations of the same underlying exponential structure. In AP Calculus BC and later in differential equations, you will see how the derivative d/dx(eˣ) = eˣ makes exponential functions the natural solutions to first-order linear ODEs, while the cycling behavior of sin and cos derivatives makes them solutions to the second-order equation y″ + y = 0.

🔮 Looking Ahead: Integration

Every derivative rule implies an antiderivative. Since d/dx(sin x) = cos x, you immediately know ∫cos x dx = sin x + C. Since d/dx(eˣ) = eˣ, you know ∫eˣ dx = eˣ + C. And since d/dx(ln x) = 1/x, you know ∫(1/x) dx = ln|x| + C. Building strong derivative fluency now directly accelerates your mastery of integration in later units of the AP Calculus AB course.

SECTION 9

Practice Problems

Which of the following statements correctly explains why the derivative of cos x is −sin x rather than sin x?

If f(x) = 4eˣ − 7 cos x + ln x, what is f′(x)?

What is the 83rd derivative of sin x?

PROBLEM 4 — APPLIED

A particle moves along the y-axis so that its position at time t > 0 is given by y(t) = 2 sin t + eᵗ − 3 ln t. (a) Find the velocity function v(t). (b) Find the acceleration function a(t). (c) Is the particle speeding up or slowing down at t = π? Justify your answer. (d) For what value(s) of t in the interval (0, π) is the velocity zero? Set up but do not solve the equation.

PROBLEM 5 — CRITICAL THINKING

Using the limit definition of the derivative and the identity cos x = sin(π/2 − x), prove that d/dx(cos x) = −sin x without directly using the cosine angle-addition formula.

SUMMARY

Lesson Summary

This lesson established the four essential transcendental derivative formulas: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, and d/dx(ln x) = 1/x. Each was derived from the limit definition of the derivative, relying on critical prerequisite limits: lim(h→0)(sin h)/h = 1 and lim(h→0)(cos h − 1)/h = 0 for the trigonometric derivatives, lim(h→0)(eʰ − 1)/h = 1 for the exponential, and the logarithmic limit identity for ln x.

Key structural insights include the period-4 cycle of successive trigonometric derivatives (useful for finding the nth derivative), the self-replicating property of eˣ (it is invariant under differentiation), and the domain restriction of ln x (x > 0). Remember that all trigonometric derivatives require radian measure, and avoid confusing the power rule (variable base, fixed exponent) with the exponential derivative (fixed base, variable exponent). These four formulas, combined with the chain rule you will learn next, will enable you to differentiate any combination of trigonometric, exponential, and logarithmic functions on the AP Calculus AB exam.