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

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.

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.

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.

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.

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.

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.

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

Derivative of cos x

Derivative of eˣ

Derivative of ln x

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.

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.

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.

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.

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.

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.