Free lesson · Differential calculus
The derivative: from an average slope to an instantaneous rate
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Start with the idea
- Average change divides the output difference by an input difference.
- The derivative is the limit of that ratio as the input difference shrinks.
- It measures local sensitivity, not the whole future path.
Symbols, units & horizon
- x: input at which slope is measured
- h: nonzero input increment before taking the limit
- f′(x), df/dx: derivative of f with respect to x
- Δf: finite output change
- lim h→0: limiting slope as the increment shrinks
- ⇒: implies for the stated function
When and why to use this
Use a derivative for instantaneous rates and local sensitivities. Open the slope lab to move the observation point and shrink the secant interval.
- A secant line joins two points on a curve.
- A tangent captures the limiting slope at one point when a finite derivative exists.
- For position as a function of time, average slope is average velocity and the derivative is instantaneous velocity.
- A derivative has output units divided by input units.
- Differentiability implies continuity, but continuity alone is not enough: the absolute-value function has a corner at zero, with left slope −1 and right slope +1.
The derivative: from an average slope to an instantaneous rate
- For f(x)=x², expand f(x+h)=(x+h)²=x²+2xh+h².
- Subtract f(x)=x² and divide by nonzero h: (2xh+h²)/h=2x+h.
- As h approaches zero, the h term disappears, leaving f′(x)=2x. At x=3 the tangent slope is 6.
At x=3, a forward step h=.1 gives slope (3.1²−3²)/.1=6.1. With h=.01 the slope is 6.01. The derivative is exactly 6.
An analogy to remember
An average journey speed uses two odometer readings. A speedometer approximates what the rate is at the present instant.
How this becomes a building block
This is a complete slope formula and also a component inside larger models. Option delta is a derivative of price with respect to spot. A cost model’s derivative is marginal cost: the extra estimated expense for a small increase in order size. Neither derivative forecasts the input change.
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 square(x): return x*x
def square_derivative(x): return 2*x
def forward_slope(function, x, h):
if h == 0: raise ValueError("Use a nonzero step")
return (function(x+h)-function(x))/h
print(square_derivative(3), forward_slope(square,3,.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
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