Math · Calculus I · Concept
The limit definition of the derivative
The derivative f′(a) is the slope of the tangent line at x = a: the limit of the secant slopes (f(a + h) − f(a))/h as h approaches 0. When that limit exists, it is the instantaneous rate of change of f at a.
Two points give an average rate
Pick two inputs, a and a + h, on the graph of f. The secant line through them has slope (f(a + h) − f(a))/h: the change in output over the change in input. That is the average rate of change of f over the interval, in output units per input unit.
Let the second point slide in
As h shrinks toward 0, the second point slides along the curve toward (a, f(a)). If the secant slopes approach one finite number from both sides, that number is the derivative f′(a), and the limiting line is the tangent line.
Why not just set h = 0?
With h = 0 the quotient is 0/0, which is undefined. The limit asks what the quotient approaches for small nonzero h, so the algebra first simplifies the quotient for h ≠ 0 and only then lets h approach 0.
Simplify before taking the limit
Expand f(a + h), subtract f(a) and factor h out of the numerator. For h ≠ 0 the factor h cancels, and the expression that remains can be evaluated at h = 0. For f(x) = x² this gives 2a at every a, so f′(x) = 2x.
The tangent line is a local model
The tangent at x = a passes through (a, f(a)) with slope f′(a). Near a it predicts small changes well; farther away the curve bends away from it. This is the idea behind linear approximation.
When there is no derivative
At a corner, such as |x| at 0, the slopes from the left (−1) and from the right (+1) disagree, so there is no single tangent slope. A vertical tangent or a jump also rules out a finite derivative. A function must be continuous at a to be differentiable there, but continuity alone is not enough.
Common mistakes
- Setting h = 0 before simplifying, which gives 0/0 instead of the derivative.
- Expanding f(a + h) as f(a) + h: for f(x) = x², (a + h)² is a² + 2ah + h², not a² + h².
- Treating one secant slope with a small nonzero h as the derivative itself: it is an estimate until the limit is taken.
- Assuming a continuous function has a derivative everywhere: |x| is continuous at 0 but has a corner there.
Key terms
- Derivative
- The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
- Difference quotient
- The average rate of change [f(a + h) − f(a)]/h between x = a and x = a + h, with h ≠ 0. Its limit as h → 0 is the derivative f′(a), when that limit exists.
- Secant line
- A line through two points on a curve; its slope is the average rate of change between them. In trigonometry, “secant” also names the function sec θ = 1/cos θ.
- Tangent line
- The line that matches a curve’s direction at a point, with slope equal to the derivative there. It can cross the curve; in trigonometry, “tangent” also names tan θ = sin θ/cos θ.
- Limit
- The value f(x) approaches as x gets closer and closer to a point a, not counting x = a itself. A limit can exist even when f(a) is undefined or has a different value.
- Differentiable
- Having a derivative at a point. A function is continuous wherever it is differentiable, but not the other way round: |x| is continuous at x = 0 with no derivative there.
Work through an example
Use the limit definition to find the slope of f(x) = x² at x = 2, then write the tangent line there.
Find a derivative from the limit definition →Sources and scope
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