Free lesson · Math & notation

Derivatives, integrals and approximations

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Start with the idea

A tangent line matches a curve’s slope at one point. A quadratic approximation also matches curvature, usually improving a small-move estimate. Neither is a guarantee far from the starting point.

Symbols, units & horizon

S: input price
V(S): value as a function of price
ΔS: finite price change, also written h
ΔV: resulting value change
V′: first derivative, value units per price unit
V″: second derivative, value units per squared price unit
≈: approximation, higher-order terms omitted
∂: partial derivative holding other inputs fixed
∫: integral, a continuous accumulation

When and why to use this

Use this as preparation for duration, option Greeks and optimisation derivatives.

A derivative measures a local rate of change. For a price function V(S), dV/dS asks how much value changes for a small change in spot S. A partial derivative ∂V/∂S changes S while holding the other inputs fixed. A second derivative measures curvature.

Δ V \approx V^{'} (S) Δ S + \frac{1}{2} V^{''} (S) (Δ S)^{2}

Elementary differentiation + Taylor expansion

Build a second-order Taylor approximation

Start with $V (S + h) = V (S) + V' (S) h + \frac{V'' (S) h^{2}}{2} + \dots$ . Subtract V(S) and write h=ΔS.
For V(S)=S², the first derivative is 2S and second derivative 2. Substitution gives $Δ V = 2 S Δ S + (Δ S)^{2}$ , exactly in this quadratic case.

Work it by hand

S=10, ΔS=1: linear estimate=20, curvature adds 1; actual change=11²−10²=21.

Δ means a finite change; ≈ means approximately equal. Prime marks denote derivatives. An integral ∫ adds a continuous stream of tiny contributions; bounds specify the interval. The symbols dS and dt describe infinitesimal changes in calculus, but dW denotes a stochastic Brownian increment in stochastic calculus and obeys different rules.

Research sources, review dates and limitations

Extend the research question

Audit units before joining two datasets. A percent, a decimal return and a currency amount cannot be combined merely because they are numbers.

Continue with the connected research module →

Connect the ideas: Cash flows and accounting

Retrieve: Track units, signed cash movements and ownership at each event.

Check the change: Instrument obligations, financing and external capital flows change the ledger you need.

Markets & returns → Trading different assets → Research & backtests → Execution & microstructure → Fund operations & capstone → Putting it all together

Explain it yourself: Why can an account balance rise without an investment profit?

Self-assessed. Write your explanation before opening this comparison.

A deposit raises the balance without being investment P&L. Reconcile external flows separately from fills, fees and marked holdings.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def taylor_change(delta_s, first_derivative, second_derivative):
    """Return a local change in value; derivatives evaluated at starting S."""
    return first_derivative*delta_s + .5*second_derivative*delta_s**2

print(taylor_change(1, 20, 2))  # V(S)=S² at S=10: 21

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