Math · Calculus · Concept
From secant to derivative
A derivative is the limiting rate seen as two nearby inputs come together.
Two points give an average
For two different inputs, the secant slope compares output change with input change. Its sign tells you whether the net change is upward or downward. The units are output units per input unit.
Approach without dividing by zero
The derivative asks whether this quotient approaches one finite value as h tends to zero from both sides. Setting h to zero in the original quotient gives 0/0, which is undefined; a limit concerns nearby inputs instead.
A local line, not a global replacement
If the derivative exists, a tangent through (a,f(a)) predicts small nearby changes. This approximation can become poor farther from a. At a corner such as |x| at zero, the two sides give different slopes, so there is no single derivative.
Common mistakes
- A secant slope at nonzero h is not necessarily the derivative.
- A tangent need not touch the curve only once.
Work through an example
Find the tangent slope for f(x)=x² at x=2.
Derive the slope of x² at 2 →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.