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
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Calculus, Vol. I (Tom Apostol, 2nd ed., 1967) foundational - Vol. I, Ch. 4 - The derivative as a limit, its geometric meaning, and differentiability vs continuity.
- Make It Stick (Brown, Roediger & McDaniel, 2014) foundational - generation effect - Deriving results from first principles as generative retrieval that builds durable understanding.