Derivatives

The idea in plain language

A derivative tells you how fast something is changing at one exact moment. Think of it as a rate of change; speed is one example.^[11] On a graph, the derivative of a function at a point is the slope of the line tangent to the graph there.^[11]

The Calculus page introduces this as the “tangent problem”. This page looks at how derivatives are actually defined, written and calculated, and where they are used.

How it works: from average to instantaneous

Pick two points on a curve and draw a straight line through them. This line is called a secant, and its slope is the average rate of change between the two points.^{[1, 11]} Now slide the second point towards the first. The tangent line is the limit of these secant lines as the distance between the two points goes to zero.^{[11, 1]}

The slope of that tangent turns up so often that it has its own name, the derivative, and the process of finding one is called differentiation.^[1] In words: as the change in x shrinks to zero, the average rate of change becomes an instantaneous rate of change.^{[11, 2]}

Motion is the clearest case. For an object whose position is s(t), the slope of a secant line is its average velocity over a time interval, and the slope of the tangent line is its instantaneous velocity.^[1] Velocity is the derivative of position, and acceleration is the derivative of velocity.^{[2, 4]}

The sign of a derivative carries meaning too. The derivative is negative where a function is decreasing, positive where it is increasing, and zero where the graph has a horizontal tangent.^[2]

The math (optional)

The definition. OpenStax writes the derivative of f at a point a as f′(a) and defines it as a limit of the difference quotient:^[1]

f′(a) = lim (x → a) [f(x) − f(a)] ÷ (x − a)

An equivalent form uses a small step h: the limit, as h → 0, of [f(a + h) − f(a)] ÷ h.^[1] Letting a vary gives the derivative function f′(x), defined for every x where that limit exists.^[2] You can’t just put h = 0 into the fraction, because you can never divide by zero; that is why a limit is needed.^[11]

Notation. Several people developed calculus, so there are several notations for the derivative.^[11] OpenStax writes one derivative both as y′ = 2x − 2 and as dy/dx = 2x − 2, and calls dy/dx Leibniz notation, common in engineering and physics.^[2] MIT’s 18.01 lecture notes label dy/dx as Leibniz’s notation and the prime form f′(x₀) as Newton’s notation.^[11] Higher derivatives are derivatives of derivatives: the second derivative of position, s″(t), is acceleration.^{[11, 4]}

The rules. Going back to the limit every time would be slow, so a few rules do most of the work:

• Constant rule: if f(x) = c, then f′(x) = 0.^[3]
• Power rule: for a positive integer n, the derivative of xⁿ is n·xⁿ⁻¹, and the result extends to polynomials.^{[3, 11]}
• Sum and constant multiple rules: the derivative of a sum is the sum of the derivatives, and a constant factor stays in front.^{[3, 11]}
• Product rule: (uv)′ = u′v + uv′.^{[11, 3]}
• Quotient rule: (u/v)′ = (u′v − uv′) ÷ v².^[11]
• Chain rule, for a function inside another function: the derivative of f(g(x)) is f′(g(x))·g′(x), or in Leibniz form dy/dx = (dy/du)·(du/dx).^{[6, 11]}

Common functions. The derivative of sin x is cos x, and the derivative of cos x is −sin x.^{[5, 11]} The exponential function eˣ is its own derivative.^[7] The derivative of the natural logarithm ln x is 1/x.^[7] For 1/x itself, MIT’s notes find the derivative −1/x₀² at a point x₀, and show that the tangent line there always cuts off a triangle of area 2 with the axes.^[11]

Worked examples (our own working, set up from the cited problems):

• For f(x) = 3x² − 4x + 1, the power, sum and constant rules give f′(x) = 6x − 4, so f′(2) = 6 × 2 − 4 = 8.^{[1, 3]}
• If f(2) = 3, f′(2) = −4, g(2) = 1 and g′(2) = 6, the product rule gives the derivative of f·g at 2 as (−4)(1) + (6)(3) = 14.^[3]
• For f(x) = x² − 4x + 6 at x = 1, f(1) = 3 and f′(x) = 2x − 4 gives a slope of −2, so the tangent line is y − 3 = −2(x − 1), or y = −2x + 5.^[3]
• A particle at position s(t) = 3t² − 4t + 1 metres has velocity s′(t) = 6t − 4 and constant acceleration s″(t) = 6 m/s².^{[2, 4]}
• By the chain rule, the derivative of sin(x³) at a point a is cos(a³)·3a².^[6]

Differentiable and continuous

If a function is differentiable at a point, it must be continuous there.^{[2, 11]} The reverse is not true: a continuous function can still fail to have a derivative.^[2] The absolute value function |x| is continuous everywhere, but its derivative at 0 is undefined.^[2] To be differentiable at a point, a function has to be “smooth” there, and the sharp corner of |x| at 0 is where its derivative is undefined.^[2]

Applications

Derivatives measure rates of change across many fields: velocity and acceleration in physics, marginal profit in business, and growth rates in biology.^{[1, 4]} Electric current, for example, is the derivative of charge with respect to time.^[11]

Estimating. For small h, f(a + h) ≈ f(a) + f′(a)·h.^[4] If f(3) = 2 and f′(3) = 5, then f(3.2) ≈ 2 + 5 × 0.2 = 3.^[4]

Motion. For a rock dropped from 64 feet, with height s(t) = −16t² + 64, the velocity is s′(t) = −32t.^{[1, 3]} One second after release that is −32 × 1 = −32 ft/s, and at t = 2 s, when s(2) = 0 and the rock lands, it is −32 × 2 = −64 ft/s.^{[1, 4]} The speed is the size of the velocity, |v(t)|, so the rock lands at 64 ft/s.^[4]

Optimisation. A common use of calculus is finding the largest or smallest value of a function.^[8] Fermat’s theorem says that if a function has a local maximum or minimum at c and is differentiable there, then f′(c) = 0.^[10] So the search looks at these critical points and at the endpoints.^[8] In OpenStax’s garden problem, 100 ft of fencing closes three sides of a rectangle against a wall.^[8] With x for each side at right angles to the wall, the area is A = x(100 − 2x), A′ = 100 − 4x is zero at x = 25, and the best garden is 25 ft by 50 ft with an area of 25 × 50 = 1,250 ft².^{[8, 10]}

Links to integrals. The Fundamental Theorem of Calculus ties derivatives to integrals: the derivative of F(x), the definite integral of f from a to x, is the original function: F′(x) = f(x), for continuous f.^[9]

Common misconceptions

A tangent line is just a line that touches the curve at one point. It is not just a line that meets the graph once; it is defined as the limit of secant lines.^[11]

The derivative of a product is the product of the derivatives. It is tempting, but the product rule is different.^[3] For x·x = x², the derivative is 2x, which is 6 at x = 3, while multiplying the two separate derivatives would give 1 × 1 = 1.^[3]

“A graph and its derivative look alike.” The graph of f(x) does not look like the graph of f′(x).^[11]

“Continuous means differentiable.” Continuity is necessary but not enough, as the corner in |x| shows.^[2]

“A zero derivative always means a maximum or minimum.” Fermat’s theorem does not say a function must have an extremum at every critical point.^[10] For x³, the derivative 3x² is 0 at x = 0, yet the function keeps increasing through that point.^{[10, 3]}

“In the chain rule, the order doesn’t matter.” Composing functions is not commutative: f(g(x)) and g(f(x)) are generally different.^[11]

History

Most mathematicians and historians agree that calculus was developed independently by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716).^[1] According to MacTutor, Newton laid its foundations while at home, several years before Leibniz’s independent discovery.^[13] He called his approach the “method of fluxions”, and it rested on his insight that integration is the inverse of differentiation.^[13] His treatise on fluxions was written in 1671 but not printed until John Colson’s English translation of 1736.^[13] The Stanford Encyclopedia of Philosophy describes how Newton later grounded fluxions in “prime and ultimate ratios”, a kinematic form of the theory of limits.^[14]

Leibniz published details of his differential calculus in 1684, in the paper Nova Methodus in the journal Acta Eruditorum; the Stanford Encyclopedia of Philosophy calls this essay the official birth of the differential calculus.^{[12, 14]} It contained the d notation and rules for derivatives of powers, products and quotients, but no proofs.^[12] Leibniz wrote dx and dy for infinitesimal differences, or differentials.^[14] MacTutor notes that he developed today’s notation but never thought of the derivative as a limit.^[12]

The priority dispute. In 1711, a paper by Keill in the Royal Society’s Transactions accused Leibniz of plagiarism.^[12] Leibniz demanded a retraction, saying he had never heard of fluxions before reading the works of Wallis.^[12] According to MacTutor, the Royal Society committee’s report, published as Commercium epistolicum in 1713, found for Newton and was written by Newton himself.^{[12, 13]} MacTutor judges the report harshly: “It was totally biased, not asking Leibniz to give his version of the events.”^[12] The Stanford Encyclopedia of Philosophy also describes the dispute as bitter.^[15] MacTutor says it took up much of Leibniz’s mathematical activity in his last years and dominated Newton’s later life.^{[12, 13]}

Making it rigorous. The infinitesimals used by the founders were controversial from the start.^[14] Berkeley aimed his criticism mainly at Newton’s fluxions.^[14] According to the Stanford Encyclopedia, the cornerstone of a rigorous calculus came from the ideas of the French mathematician Augustin-Louis Cauchy (1789–1857).^[14]

Going further

Next on the map is Integrals, the other half of calculus. The free OpenStax Calculus Volume 1 textbook and MIT’s single-variable calculus course, listed below, cover all of this with many more worked examples.