Teaching Notes · Math for Social Sciences

The Derivative and What It Measures

Everything in the previous set of notes was building toward one idea: a precise way to describe how fast a function is changing at a single instant, not on average over some interval, but exactly at one point. That idea is the derivative, and in a business course it earns its keep almost immediately, since marginal cost, marginal revenue, and marginal profit are nothing more than derivatives wearing economic names.

The derivative, defined

The derivative of f(x) at x is given by the limit

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

when that limit exists, and a function with a derivative at a point is called differentiable there. This looks abstract until it is applied to an actual function. For f(x) = 2x + 5, expanding the definition gives [2(x+h)+5 − (2x+5)]/h = 2h/h = 2, so f'(x) = 2, a constant, which makes sense since the original function is already a straight line of slope 2. For f(x) = 3x² − 2x, the same process gives, after expanding (x+h)² and cancelling the terms that do not involve h, f'(x) = 6x − 2, a function that now depends on x, meaning the rate of change of 3x² − 2x is itself different at every point.

What the derivative measures geometrically

Picture two points on the graph of f: P at (x, f(x)) and Q at (x+h, f(x+h)). The line joining them is a secant line, with slope [f(x+h) − f(x)]/h. As h shrinks toward zero, Q slides along the curve toward P, and the secant line settles into the tangent line at P. The derivative is exactly the slope of that tangent line, the slope the curve is following at that single instant.

For f(x) = x² + 1, working through the definition gives a slope formula of f'(x) = 2x. At the point (−1, 2), the slope is 2(−1) = −2; at (2, 5), the slope is 2(2) = 4, two different tangent slopes along the same curve, each read straight from the one formula. One further fact is worth keeping in mind going forward: if a function is differentiable at a point, it is automatically continuous there. The reverse does not hold, a function can be continuous at a point and still fail to have a derivative there, but a function that is discontinuous at a point can never have a derivative at that point either.

Shortcuts: the rules of differentiation

Working from the limit definition every time would make calculus unworkable in practice, so a short list of rules covers almost everything that follows. The derivative of a constant is always zero. The power rule says that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹, so the derivative of x⁴ is 4x³. A constant multiplying a function simply carries through to the derivative unchanged, so the derivative of 4x⁷ is 4(7x⁶) = 28x⁶. And the derivative of a sum is the sum of the derivatives, so for f(x) = 3x⁴ + 5x⁻³, the derivative is 3(4x³) + 5(−3x⁻⁴) = 12x³ − 15x⁻⁴.

These rules make tangent lines quick to find. For f(x) = x² − 2 at the point (1, −1), the derivative is f'(x) = 2x, so the slope at that point is 2(1) = 2, giving the tangent line y + 1 = 2(x − 1), or y = 2x − 3. For f(x) = x³ + x at (−1, −2), the derivative is f'(x) = 3x² + 1, giving a slope of 3(−1)² + 1 = 4 at that point, and a tangent line of y = 4x + 2.

Rate of change, and the marginals of economics

Away from geometry, f'(x) is simply the rate of change of f with respect to x. A revenue model for a corporation, built from data over several years, is given by R = 174.343t³ − 5630.45t² + 63,029.8t − 218,635, where t = 8 represents the first year of the model. Differentiating gives the rate of change of revenue, dR/dt = 523.029t² − 11,260.90t + 63,029. Evaluated at t = 9, the second year of the model, this rate works out to about 4047, meaning revenue was increasing at roughly 4047 (in millions of dollars) per year at that point.

In economics this same idea gets specific names. If C, R, and P denote the cost, revenue, and profit of producing and selling x units, with P = R − C, then dC/dx is the marginal cost, dR/dx is the marginal revenue, and dP/dx is the marginal profit, each one the rate at which that quantity is changing per additional unit. For a profit function P = 0.0002x³ + 10x, the marginal profit is dP/dx = 0.0006x² + 10, and at a production level of 50 units this works out to 0.0006(2500) + 10 = 11.50, meaning the fiftieth unit adds roughly 11.50 to profit.

Revenue itself usually starts from a demand function, the relationship p = f(x) between the price p a business can charge and the quantity x it can expect to sell at that price, since total revenue is always R = xp. A restaurant chain estimates monthly demand for its hamburgers as p = (60,000 − x)/20,000, so revenue is R = xp = (60,000x − x²)/20,000. The marginal revenue is dR/dx = (60,000 − 2x)/20,000, and at x = 20,000 units this equals exactly 1, meaning one additional hamburger sold at that volume adds about a dollar to revenue.

Extending the same example with a cost function, C = 5000 + 0.56x, gives a profit of P = 2.44x − x²/20,000 − 5000 and a marginal profit of dP/dx = 2.44 − x/10,000. Checking three production levels shows how the margin itself changes with scale: at 20,000 units, profit is 23,800 and marginal profit is 0.44 per unit, still rising; at 24,400 units, profit peaks at 24,768 and marginal profit falls to exactly 0; at 30,000 units, profit has fallen back to 23,200 and marginal profit has gone negative, −0.56 per unit. The marginal profit crossing zero at 24,400 units is not a coincidence, it is exactly the production level where profit stops rising and starts falling, a connection the next set of notes develops into a full method for finding the best possible outcome.

Differentiating products, quotients, and compositions

Many of the functions that actually appear in business models are built out of simpler pieces multiplied, divided, or nested inside one another, and three further rules cover those cases. The product rule states that for two differentiable functions f and g, the derivative of their product is f'(x)g(x) + f(x)g'(x), not simply f'(x)g'(x). For y = (3x − 2x²)(5 + 4x), applying the product rule and simplifying gives y' = 15 + 4x − 24x². The quotient rule handles a ratio of two functions: the derivative of f(x)/g(x) is [f'(x)g(x) − f(x)g'(x)] / [g(x)]². For y = (x² + 2x)/(3x − 5), this works out to y' = (3x² − 10x − 10)/(3x − 5)².

The chain rule applies when one function is nested inside another, y = f(u) where u = g(x) is itself a function of x, in which case dy/dx = (dy/du)·(du/dx). A direct consequence, called the general power rule, covers any expression of the form y = [u(x)]ⁿ: its derivative is n[u(x)]ⁿ⁻¹·u'(x). For f(x) = (3x − 2x²)³, this gives f'(x) = 3(3x − 2x²)²(3 − 4x). The same rule finds a tangent line to a curve built from a fractional power: for y = [(x² + 4)²]^1/3, the derivative simplifies to y' = 4x / [3·∛(x² + 4)], and at x = 2 this gives a slope of 4/3 at the point (2, 4), for a tangent line of y = (4/3)x + 4/3.

Higher order derivatives

Since the derivative of a function is itself a function, it can be differentiated again. The derivative of f'(x) is called the second derivative, written f''(x), and differentiating once more gives the third derivative, f'''(x), and so on for as many orders as needed. For f(x) = 2x⁴ − 3x², the first three derivatives are f'(x) = 8x³ − 6x, f''(x) = 24x² − 6, and f'''(x) = 48x. The second derivative in particular will turn out to matter a great deal in the next set of notes, since it is what finally distinguishes a maximum from a minimum.

Source: lecture notes for an introductory calculus course for the social sciences, covering the definition of the derivative, its geometric meaning, the rules of differentiation, equations of tangent lines, rate of change, marginal cost, revenue, and profit, the product, quotient, and chain rules, and higher order derivatives.