The Intuitive Concept of a Derivative

Write $x_2=x_1+h$ and let $h$ approach zero through nonzero values. The nearby secants illustrate the search for one limiting slope at the fixed base point.

Conditions: $x_2$ is expressed as $x_1+h$.; $h$ approaches zero through nonzero values.

For two distinct inputs $x_1$ and $x_2$, the secant slope is $\frac{f(x_2)-f(x_1)}{x_2-x_1}$. It compares the change in output with the change in input.

Conditions: The inputs $x_1$ and $x_2$ are distinct.; The function $f$ is defined at both $x_1$ and $x_2$.

The secant gives an average rate across an interval. The derivative, where it exists, gives the instantaneous rate at the point and the corresponding tangent slope.

Conditions: The derivative must exist at the point for the instantaneous rate to be defined.

When the finite limit exists, $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. We do not substitute $h=0$ directly into the quotient; we evaluate its limit.

Conditions: The finite limit must exist.; $h$ approaches zero.