Difficulty: Intermediate. Prerequisites: Limits and Continuity (Part 1 of these notes).
The derivative measures how fast a function's output changes as its input changes, at a single instant. It is defined as the limit of the difference quotient as h approaches 0. Shortcut rules (power rule, constant rule, sum rule) let you differentiate most functions without computing that limit every time. In business calculus, the derivative of a cost, revenue, or profit function gives the marginal cost, revenue, or profit, which estimates the effect of producing one more unit.
Derivative
The instantaneous rate of change of f(x) with respect to x. Formally: f'(x) = lim(h→0) [f(x+h) − f(x)] / h, provided the limit exists.
In simple terms, this means: "How fast is y changing at this exact instant?"
Differentiable
A function is differentiable at a point if its derivative exists there. If f'(x) exists, f is said to be differentiable at x.
Think of it as: "The function is smooth enough at that point to have a well-defined slope."
Difference quotient
The expression [f(a+h) − f(a)] / h, which gives the average rate of change of f over the interval from a to a+h. It is also the slope of the secant line through those two points.
Think of it as: "The slope formula applied to a curve over a small interval."
Average rate of change
The change in f(x) divided by the change in x over an interval. Equals the slope of the secant line connecting two points on the graph.
In simple terms, this means: "Your average speed over a whole trip."
Instantaneous rate of change
The rate of change at a single moment. Found by taking the limit of the difference quotient as h → 0. Equals the slope of the tangent line at that point.
In simple terms, this means: "Your speedometer reading at one instant."
Secant line
A line that passes through two points on a curve. Its slope equals the average rate of change between those points.
Tangent line
A line that touches the curve at exactly one point (locally) and has the same slope as the curve at that point. Its slope equals the derivative, the instantaneous rate of change.
Marginal cost / marginal revenue / marginal profit
The derivative of the total cost, revenue, or profit function, respectively. C'(x) approximates the cost of producing the (x+1)th unit. R'(x) approximates the revenue from selling the (x+1)th unit. P'(x) = R'(x) − C'(x).
Think of it as: "What does one more unit cost you, earn you, or net you?"
Natural exponential function
The function y = e^x, where e ≈ 2.71828. Defined by the limit e = lim(n→∞) (1 + 1/n)^n. Its key property: d/dx(e^x) = e^x. The function is its own derivative.
Natural logarithm (ln x)
The inverse of e^x. Defined for x > 0. Its derivative is d/dx(ln x) = 1/x.
Think of it as: "The power you raise e to in order to get x."
Average rate of change from x = a to x = a + h: [f(a+h) − f(a)] / h. This is the slope of the secant line.
Instantaneous rate of change at x = a: lim(h→0) [f(a+h) − f(a)] / h. This is the slope of the tangent line.
The word "rate of change" on its own always means the instantaneous rate of change.
Example: f(x) = 3x² − 1. Average rate of change from x = 2 to x = 5: [f(5) − f(2)] / (5 − 2) = [74 − 11] / 3 = 21. Slope of the tangent line at x = 2: f'(x) = 6x, so f'(2) = 12.
f'(x) = lim(h→0) [f(x+h) − f(x)] / h, if the limit exists.
The process of finding the derivative is called differentiation.
Key identity: derivative = instantaneous rate of change = slope of the tangent line. These three phrases all mean the same thing.
Notation: f'(x), y', dy/dx all denote the derivative.
Constant rule: d/dx[c] = 0. The derivative of any constant is zero.
Power rule: d/dx[x^n] = n·x^(n−1). Works for any real number n, including negative and fractional exponents.
Constant multiple rule: d/dx[c·f(x)] = c·f'(x). Constants factor out.
Sum/difference rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x). Differentiate term by term.
Applying the rules: Before differentiating, rewrite any roots, fractions, or denominators involving x as power expressions.
1/x = x^(−1), so d/dx[1/x] = −x^(−2) = −1/x².
√x = x^(1/2), so d/dx[√x] = (1/2)x^(−1/2) = 1/(2√x).
5/x³ = 5x^(−3), so d/dx[5/x³] = −15x^(−4) = −15/x⁴.
Example: y = 5x³ − (1/2)x² + 12x − 8. Then y' = 15x² − x + 12.
d/dx[e^x] = e^x. The natural exponential is its own derivative.
d/dx[e^(Cx)] = C·e^(Cx), where C is a constant.
d/dx[ln x] = 1/x, for x > 0.
Use log properties to simplify before differentiating: ln(x³) = 3 ln x, so d/dx[ln(x³)] = 3/x.
Example: f(x) = 13x³ − 5√x + 50 ln x. Then f'(x) = 39x² − 5/(2√x) + 50/x.
e = lim(n→∞) (1 + 1/n)^n ≈ 2.71828.
Compound interest: A = P(1 + r/m)^(mt), where P = principal, r = annual rate, m = compounding periods per year, t = years.
Continuous compounding: A = Pe^(rt). This is the limit of the compound interest formula as m → ∞.
To find how long to reach a target amount with continuous compounding, set Pe^(rt) = target, divide by P, take ln of both sides, solve for t.
Example: How long for $10,000 to grow to $15,000 at 9% compounded continuously? 15000 = 10000e^(0.09t). Then 1.5 = e^(0.09t), ln(1.5) = 0.09t, t = ln(1.5)/0.09 ≈ 4.51 years.
Profit = Revenue − Cost: P(x) = R(x) − C(x).
Revenue = quantity × price: R(x) = x·p(x), where p(x) is the price-demand function.
Marginal cost C'(x) approximates the cost of the (x+1)th unit.
Marginal revenue R'(x) approximates the revenue from selling the (x+1)th unit.
Marginal profit P'(x) = R'(x) − C'(x).
Average cost = C(x)/x. Marginal average cost = d/dx[C(x)/x]. Note: find the average function first, then differentiate.
Example: C(q) = 0.001q³ − 0.05q² + 40q + 4000. C'(q) = 0.003q² − 0.1q + 40. Marginal cost at q = 250: C'(250) = 0.003(62500) − 0.1(250) + 40 = 187.5 − 25 + 40 = $202.50. This estimates the cost of producing the 251st unit.
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[x^n] = nx^{n-1}\frac{d}{dx}[cf(x)] = c\,f'(x) \qquad \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[e^{Cx}] = Ce^{Cx} \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}A = P\left(1 + \frac{r}{m}\right)^{mt} \quad \text{(compound interest)}A = Pe^{rt} \quad \text{(continuous compounding)}P(x) = R(x) - C(x) \qquad R(x) = x \cdot p(x)\bar{C}(x) = \frac{C(x)}{x} \quad \text{(average cost)}The derivative appears whenever you need to know how fast something is changing right now: the rate at which a population is growing at a given moment, the rate at which revenue is increasing at a production level of 5,000 units, or the rate at which an investment balance is growing at a specific time. Marginal analysis is one of the most direct business applications: if C'(250) = $202.50, that tells a production manager roughly what the 251st unit will cost.
Students often confuse average rate of change with instantaneous rate of change. Average rate uses two points (secant line). Instantaneous rate uses one point (tangent line, via the limit).
Students often forget to rewrite expressions before differentiating. You cannot apply the power rule to 5/x³ as written. Rewrite it as 5x^(−3) first.
Students often think the derivative of e^(3x) is e^(3x). The constant in the exponent must come out front: the answer is 3e^(3x).
Students often confuse marginal cost with average cost. Marginal cost C'(x) is the derivative of total cost. Average cost is C(x)/x. These are different functions with different meanings.
⚠️ Power rule with negative and fractional exponents: expect to differentiate expressions like 1/√x or 2/x⁵. Rewrite, then apply the rule.
⚠️ Tangent line equations: given f(x) and a point x = a, find f(a) for the y-coordinate, f'(a) for the slope, then use point-slope form y − f(a) = f'(a)(x − a).
⚠️ Horizontal tangent lines: set f'(x) = 0 and solve for x. These are the x-values where the tangent line is flat.
⚠️ Marginal analysis: C'(x) at x = n approximates the cost of unit n+1. Exam questions will ask you to compare the marginal estimate to the actual cost C(n+1) − C(n).
⚠️ Continuous compounding word problems: set up A = Pe^(rt), isolate the unknown, use ln to solve.
True or false: The derivative of a constant is 1.
Fill in the blank: d/dx[x⁷] = __________.
True or false: d/dx[e^(5x)] = e^(5x).
Fill in the blank: If C(x) is total cost, then C'(x) is the __________ function.
True or false: To find where the tangent line is horizontal, set f'(x) = 0.
Answers: 1. False (it is 0). 2. 7x⁶. 3. False (it is 5e^(5x)). 4. Marginal cost. 5. True.
Q: Differentiate f(x) = 1/x + 7.
A: Rewrite as x^(−1) + 7. Then f'(x) = −x^(−2) + 0 = −1/x².
Q: Find the equation of the tangent line to f(x) = x⁴ − 8x³ + 7 at x = 1.
A: f(1) = 1 − 8 + 7 = 0. f'(x) = 4x³ − 24x², so f'(1) = 4 − 24 = −20. Tangent line: y − 0 = −20(x − 1), or y = −20x + 20.
Q: A company's sales function is S(t) = 0.04t³ + 0.6t² + t + 5 (millions of dollars, t in months). Find S'(t) and interpret S(5) and S'(5).
A: S'(t) = 0.12t² + 1.2t + 1. S(5) = 0.04(125) + 0.6(25) + 5 + 5 = 5 + 15 + 5 + 5 = 30 million. S'(5) = 0.12(25) + 1.2(5) + 1 = 3 + 6 + 1 = 10 million per month. After 5 months, total sales are $30 million and sales are increasing at $10 million per month.
Q: How long will it take money to triple at 5.5% compounded continuously?
A: 3P = Pe^(0.055t). Divide: 3 = e^(0.055t). Take ln: ln 3 = 0.055t. Then t = ln(3)/0.055 ≈ 19.97 years.
Q: Find the marginal cost and marginal revenue given C(x) = (1/4)x² + 3x + 67 and p(x) = (1/5)(45 − x).
A: C'(x) = (1/2)x + 3. R(x) = x·p(x) = (1/5)(45x − x²) = 9x − (1/5)x². R'(x) = 9 − (2/5)x.
The derivative definition builds directly on limits (Part 1). The product rule, quotient rule, and chain rule (Part 3) extend these basic rules to more complex functions. Marginal analysis connects to optimisation (Part 4), where you set the derivative to zero to find maximum profit or minimum cost. The antiderivative (Part 4) reverses differentiation and leads to integration.
Tags: derivative, differentiation, power rule, constant rule, sum rule, difference quotient, average rate of change, instantaneous rate of change, secant line, tangent line, horizontal tangent, marginal cost, marginal revenue, marginal profit, average cost, natural exponential, e^x, ln x, compound interest, continuous compounding, MAC 2233, business calculus, FIU