From One Rate to a Function of Rates
A derivative is often introduced as a number: the instantaneous rate of change of a function at a particular input. But the input itself can vary. At one input the rate may have one value; at another input it may have a different value. Recording the rate at every input where it exists creates a new function.
This distinction is useful because a derivative is not generally one fixed number. The notation \(f'(x)\) describes a rule whose input is \(x\). By contrast, \(f'(a)\) is the particular output of that rule when its input is the specified number \(a\).
Definition and Domain
Let \(f:I\to\mathbb{R}\), where \(I\) is an open interval. At an input \(x\in I\), the derivative is defined by the limit of the difference quotient, if that limit exists as a finite real number:
Here \(h\) is the change in input. In taking the limit, \(x\) is held fixed and \(h\) approaches zero. The value of the limit may depend on which \(x\) was held fixed.
The domain of the derivative function need not be all of \(I\). If the derivative limit fails at some input, \(f'\) has no value there. This does not prevent \(f'\) from being a function: a function can have a domain that is smaller than the domain of the original function.
It helps to keep the roles of the two functions separate. The original function \(f\) assigns an output \(f(x)\) to an input \(x\). The derivative function \(f'\) assigns a rate \(f'(x)\) to that same input, where the derivative exists.
| Notation | What it denotes |
|---|---|
| \(f\) | The original function |
| \(f(x)\) | The output of the original function at input \(x\) |
| \(f'\) | The derivative function |
| \(f'(x)\) | The output of the derivative function at input \(x\) |
Worked Example: Deriving the Function for a Square
Worked Example: Deriving the Function for a Square
Let \(f:\mathbb{R}\to\mathbb{R}\) be \(f(x)=x^2\). Find the derivative function from its definition.
Begin with an arbitrary fixed input \(x\), rather than choosing one numerical input. For \(h\ne 0\), the difference quotient is
Now take the limit as \(h\to 0\), with \(x\) fixed:
The calculation works for every real \(x\), so the derivative function is \(f':\mathbb{R}\to\mathbb{R}\), \(f'(x)=2x\). As a check, substituting any fixed input \(a\) into the original difference quotient gives \(2a+h\), whose limit is \(2a\), agreeing with the formula \(f'(a)=2a\).
The expression \(2x\) is not one derivative value. It is the rule for producing a derivative value from an input. For example, substituting \(x=3\) into this rule gives \(f'(3)=6\). The rule \(f'(x)=2x\) describes all such outputs, not just the one obtained at \(3\).
Worked Example: A Linear Function Has a Constant Derivative Function
Worked Example: A Linear Function Has a Constant Derivative Function
Let \(g:\mathbb{R}\to\mathbb{R}\) be \(g(x)=4x-3\). Find \(g'\) using the difference quotient.
For an arbitrary fixed \(x\), first compute \(g(x+h)\):
Then form and simplify the quotient:
Taking the limit gives
This holds for every \(x\in\mathbb{R}\), so \(g'\) is the constant function \(g':\mathbb{R}\to\mathbb{R}\) with \(g'(x)=4\). To check, substituting the result into the difference quotient gives the same value \(4\) for every nonzero \(h\), so its limit is indeed \(4\) at every input.
This example shows that the derivative function can happen to be constant. That is a particular outcome, not the meaning of derivative function in general. In the square-function example, the derivative depends on \(x\); here it does not.
Why the Derivative Formula Depends on the Input
The derivative is defined point by point. At each fixed \(x\), we examine the difference quotient near that input and take its limit. Repeating that process for all eligible inputs gives the derivative function. When the resulting limits vary with \(x\), the derivative function records that variation.
Keep \(x\) symbolic so the calculation can apply across the domain, not just at one selected point.
Compare \(f(x+h)\) with \(f(x)\), divide by the nonzero input change \(h\), and simplify.
If a finite limit exists, its value is the derivative at that fixed \(x\).
The rule obtained as \(x\) varies is the derivative function on the set where those limits exist.
The method works because the definition specifies exactly how to find the derivative at each input. Keeping \(x\) arbitrary allows one calculation to describe the result at every input for which the limit exists. If one substitutes a number for \(x\) at the beginning, the calculation instead addresses only that one input.
Worked Example: A Cubic Produces a Quadratic Derivative Function
Worked Example: A Cubic Produces a Quadratic Derivative Function
Let \(p:\mathbb{R}\to\mathbb{R}\) be \(p(x)=x^3-2x\). Find its derivative function directly from the definition.
For fixed \(x\), evaluate \(p(x+h)\):
Subtract \(p(x)=x^3-2x\) and divide by \(h\ne 0\):
Now let \(h\to 0\):
Thus \(p':\mathbb{R}\to\mathbb{R}\) is given by \(p'(x)=3x^2-2\). To check the result, subtract this proposed limit from the simplified quotient:
For fixed \(x\), \(3xh+h^2\to 0\) as \(h\to 0\). Therefore the quotient approaches \(3x^2-2\), as claimed.
A Short Proof That the Derivative Assignment Is Unambiguous
At any fixed input, the difference quotient has at most one limit. This guarantees that the derivative definition assigns no more than one output to each input.
A Common Mistake: Fixing the Input Too Soon
Suppose \(f(x)=x^2\), and the goal is to find the derivative function. A common mistake is to choose \(x=2\) before completing the calculation, then conclude that the derivative is \(4\). That calculation can find \(f'(2)\), but it cannot establish the derivative function at every input.
The correction is to keep \(x\) arbitrary until after taking the limit:
Only after obtaining the full rule should a particular input be substituted. Another related error is to assume that \(f'\) must have exactly the same domain as \(f\). The derivative function is defined only at inputs where the derivative limit exists.
Why the Derivative Function Matters
Thinking of \(f'\) as a function makes it possible to describe how rates vary across inputs. A calculation at one input answers a local question; the derivative function gathers the answers into one rule. That rule can depend on its input, as \(2x\) and \(3x^2-2\) do, or it can be constant, as \(4\) is.
When working with derivatives, first be clear about which object is being requested. If the task asks for a derivative function, preserve the variable \(x\) and state the domain where the derivative exists. If it asks for the derivative at a specified input \(a\), evaluate the derivative there to obtain the single number \(f'(a)\). The distinction prevents a pointwise calculation from being mistaken for a general rule.
Check Your Understanding
Answer these questions using the definitions and examples above.
- In the definition of \(f'(x)\), which quantity is fixed while the limit is taken, and which quantity approaches zero?
- For \(f(x)=x^2\), explain the difference between the statements \(f'(x)=2x\) and \(f'(3)=6\).
- Can a derivative function be constant? Give an example from this tutorial and state its derivative function.
- What determines the domain of \(f'\)?
- Why does computing the difference quotient at one numerical input not, by itself, find the derivative function?
