Free lesson · Differential calculus
Second derivatives, Taylor expansions and numerical error
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Start with the idea
- The first derivative tells you the local slope.
- The second derivative tells you how that slope changes.
- Adding curvature to a tangent approximation explains why a straight-line estimate misses curved behaviour.
Symbols, units & horizon
- f′,f″: first and second derivatives
- h: input change or finite-difference step
- R₃: omitted third-and-higher-order contribution
- M: bound on absolute third derivative for the stated error bound
- |h|: step magnitude
- ≈: approximate equality
- 2h: distance between the two symmetric sample points
When and why to use this
Use curvature to improve a local approximation and to understand when numerical derivatives are trustworthy. Compare several step sizes, especially when evaluating a noisy simulation.
- A second derivative is obtained by differentiating the first derivative.
- A positive second derivative describes local upward curvature; it does not necessarily mean the function itself is increasing.
- For example, x² is convex even where its slope is negative.
- A Taylor expansion is a local approximation with a remainder.
- If a third derivative is bounded by M across the relevant interval, the second-order remainder is bounded by M|h|³/6.
- Without such regularity or a sufficiently small step, the approximation may be poor.
Second derivatives, Taylor expansions and numerical error
- Match the value, slope and curvature with a quadratic in h: coefficients are f(x), f′(x) and f″(x)/2 because the second derivative of h² is 2.
- Write Taylor expansions at +h and −h. Subtracting cancels f(x) and the even powers. Dividing by 2h gives the central derivative formula with an error of order h² for a sufficiently smooth function.
- For f(x)=x³ at x=2, f=8, f′=12, f″=12. At h=.1 the quadratic estimate is 8+1.2+.06=9.26; the omitted cubic is .001.
The exact value 2.1³ is 9.261. The central slope at x=2 with h=.1 is (2.1³−1.9³)/.2=12.01, compared with the exact derivative 12.
An analogy to remember
A tangent is a straight ruler touching a bend in a road. A quadratic approximation also matches how tightly the road bends, but neither tells you the route far away.
How this becomes a building block
Delta and gamma are the first two spot derivatives in an option P&L approximation. Duration and convexity play a related role for bond price versus yield. Taylor expansions also explain truncation error in simulation and finite-difference solvers.
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 taylor2(value, first, second, step):
return value+first*step+.5*second*step**2
def central_derivative(function,x,h):
if h<=0: raise ValueError("Positive step required")
return (function(x+h)-function(x-h))/(2*h)
def central_second(function,x,h):
if h<=0: raise ValueError("Positive step required")
return (function(x+h)-2*function(x)+function(x-h))/h**2
print(taylor2(8,12,12,.1),central_derivative(lambda x:x**3,2,.1))Continue learning
Differential Calculus: Limits & Derivatives — all lessons- Start with a straight line: input, output and slope
- Functions, limits and continuity
- The derivative: from an average slope to an instantaneous rate
- Core derivative rules: constants, powers and sums
- Derivative rules: powers, products, quotients and compositions
- Second derivatives, Taylor expansions and numerical error
- Stationary points, optimisation and Newton’s method
Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations