Textbook
For a real function defined in a neighborhood of $x$, the derivative is the following finite limit, when it exists:
$$f'(x)=\frac{df}{dx}(x):=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.$$
The denominator h is nonzero when forming the quotient. The quotient measures average output change per unit input change; its limit gives the local rate. The notation df/dx denotes the derivative as a whole, and rules for manipulating it require mathematical justification.
For $f(x)=x^2$, the difference is $2xh+h^2$, so the quotient is $2x+h$ and the derivative is $2x$. At 3, the linear part of the change is $6h$ and the exact change is $6h+h^2$. The omitted remainder divided by $|h|$ tends to zero.
Derivatives need not exist everywhere. For $f(x)=|x|$ at zero, quotients from the left tend to −1 and from the right to 1, preventing a common limit. The second derivative $f^{\prime\prime}(x)$ or $d^2f/dx^2$ differentiates the first derivative when possible.
A two-sided limit requires consistent behaviour for both positive and negative $h$. For the square function, $2x+h$ tends to $2x$ from either direction. Once the derivative is known, writing $f(x+h)=f(x)+f'(x)h+r(h)$ means $r(h)/h\to0$. The derivative is therefore the coefficient of a local linear approximation, rather than the exact output change for an arbitrary finite step.
Definitions and notation: Basic Analysis I.
Related concepts: Function limits: making “approaches” precise.
In plain language
A derivative resembles a speedometer reading at a particular moment rather than the average speed over a whole journey. It describes the limiting ratio of changes as the step shrinks from both sides of the point. For the square function at 3, the local slope is 6 and the linear prediction’s error equals the squared step. A corner may have no common slope; a finer ruler cannot turn a table corner into a smooth bend.
Example
The constructed function is $x^2$ at 3. The example shrinks a positive step $h$ and compares the difference quotient with derivative 6.
Derivatives: local rates of change
Step 1
For h = 1 the exact change and quotient both equal 7.
Step 1 of 4: Step 1
Read all steps
Step 1
For h = 1 the exact change and quotient both equal 7.
- h; change; quotient
- 1current77current
Step 0.1
The quotient becomes 6.1; linear-approximation error is 0.01.
- h; change; quotient
- 0.1current0.616.1current
Step 0.01
The quotient becomes 6.01; linear-approximation error is 0.0001.
- h; change; quotient
- 0.01current0.06016.01included
Limit of the quotient
For every nonzero step the quotient is 6 + h, so its limit is 6.
- Linear part; remainder
- 6hincludedh²
- Derivative
- 6included
The table contains two different errors: the difference quotient differs from 6 by $h$, whereas the exact change differs from the linear term $6h$ by $h^2$. For step 0.1, these are 0.1 and 0.01 respectively. Negative steps would give quotients below 6 but approach the same limit. Substituting $h=0$ into the original fraction is forbidden; the limit uses nonzero steps.
A derivative supplies a local model. At a finite step, the remainder matters: an approximation is not exact.