The Derivative: Definition and Meaning
The derivative as the limit of secant slopes, its interpretation as instantaneous rate of change, and the marginal reasoning that pervades finance.
Leads to: Differentiation rules (2.3), Taylor approximation (2.4) and optimization (2.5) all build directly on this definition.
Learning Objectives
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- Define the derivative as a limit of difference quotients and compute it from first principles.
- Interpret the derivative geometrically (tangent slope) and physically (instantaneous rate).
- Explain why differentiability implies continuity but not conversely.
- Apply marginal reasoning and compute the growth rate of a continuously compounded balance.
Key Vocabulary
- Difference quotient
- (f(x+h)−f(x))/h, the average rate of change / secant slope over a step h.
- Derivative
- f′(x)=lim_{h→0}(f(x+h)−f(x))/h when the limit exists; the instantaneous rate.
- Tangent line
- The line through (x,f(x)) with slope f′(x); the best local linear fit to the graph.
- Differentiable
- Having a derivative at the point; requires the secant slopes to converge from both sides.
- Marginal quantity
- In economics/finance, the derivative of a total with respect to quantity (marginal cost, marginal utility).
- Continuous compounding
- Growth governed by dA/dt=rA, giving A(t)=A_0 e^{rt}.
From average to instantaneous rate
Over a step of size \(h\), the average rate of change of \(f\) is the difference quotient - the slope of the secant line through \((x,f(x))\) and \((x+h,f(x+h))\). Shrinking \(h\to0\) turns the secant into the tangent and defines the derivative:
Two readings of one number
Geometrically \(f'(x)\) is the slope of the tangent. Physically it is an instantaneous rate: if \(s(t)\) is position then \(s'(t)\) is velocity. In finance the same idea is marginal reasoning - marginal cost is the derivative of total cost, and an option’s delta is the derivative of its price with respect to the underlying.
Finance link: continuous compounding
A balance earning interest at annual rate \(r\), compounded continuously, obeys \(\frac{dA}{dt}=rA\) - its growth rate is proportional to its size. The solution is \(A(t)=A_0e^{rt}\), and differentiating confirms \(A'(t)=rA_0e^{rt}=rA(t)\). The derivative is the instantaneous interest accrual.
Interactive: secant → tangent
- Cancelling the h in the difference quotient before recognizing you may only do so for h≠0 (which is fine inside a limit).
- Claiming continuity implies differentiability; |x| at 0 is the standing counterexample.
- Confusing the average rate (secant, finite h) with the instantaneous rate (tangent, h→0).
- Writing dA/dt=r instead of dA/dt=rA for continuous compounding - growth is proportional to the balance.
- First-principles derivatives are graded on the algebra: expand, cancel the lone h, then take the limit.
- Read f′ in the units of the problem: dollars per unit (marginal cost), meters per second (velocity), 1/year (growth rate).
- A kink or vertical tangent signals non-differentiability even where the function is continuous.
- e^{rt} is the unique growth law whose derivative is a constant multiple of itself - that is why it dominates finance.
Knowledge Check
Practical Exercise
Using only the limit definition, compute \(f'(x)\) for \(f(x)=1/x\) (with \(x\ne0\)). Then state the tangent-line slope at \(x=2\) and interpret its sign.
Form the difference quotient and simplify:
Let \(h\to0\): \(f'(x)=-1/x^2\). At \(x=2\) the slope is \(-1/4\). It is negative because \(1/x\) is decreasing for \(x\gt 0\) - the tangent slopes downward.
Lesson Summary
Formula Sheet Additions
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: f′(x)=lim_{h→0}(f(x+h)−f(x))/h; it is the slope of the tangent line and the instantaneous rate of change (velocity, marginal cost, growth rate).
A: No; |x| is continuous at 0 but has a corner where left and right secant slopes disagree (−1 vs +1), so no derivative exists there. The valid implication runs the other way.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check