DIFFERENTIATION 02
Derivative Notation
Several notations describe the derivative. Each expresses the same underlying idea: how one quantity changes with respect to another.
The main idea
Suppose
\[ y=f(x). \]The derivative may be written in several equivalent ways:
These notations describe the same derivative, but each emphasises something slightly different.
1. Prime notation
If
\[ y=f(x), \]then the derivative can be written
\[ f'(x). \]This is read as
“\(f\) prime of \(x\)”.
For example, if
\[ f(x)=x^2, \]then
\[ f'(x)=2x. \]Prime notation is compact and convenient when working directly with functions.
2. \(y'\) notation
If the function is written as
\[ y=f(x), \]we may also write its derivative as
\[ y'. \]So if
\[ y=x^2, \]then
\[ y'=2x. \]This is simply a shorter form of derivative notation.
3. Leibniz notation
Another important notation is
This is read as
“the derivative of \(y\) with respect to \(x\)”.
It emphasises which quantity is changing and which variable the change is measured against.
4. Why \(dy/dx\) looks like a fraction
Before taking a limit, the average rate of change is
\[ \frac{\Delta y}{\Delta x}. \]The derivative is obtained from the limit
\[ \frac{dy}{dx} = \lim_{\Delta x\to0} \frac{\Delta y}{\Delta x}. \]So the notation \(dy/dx\) reflects the idea of change in \(y\) relative to change in \(x\).
5. Is \(dy/dx\) literally a fraction?
Not in its original definition.
The expression
\[ \frac{\Delta y}{\Delta x} \]is an ordinary quotient of two finite changes.
But
\[ \frac{dy}{dx} \]denotes the derivative obtained through a limiting process.
Later, differentials \(dx\) and \(dy\) will be defined carefully. This explains why some fraction-like manipulations with \(dy/dx\) are valid.
6. Operator notation
The expression
\[ \frac{d}{dx} \]can be viewed as a differentiation operator.
It means:
“differentiate with respect to \(x\)”.
For example,
\[ \frac{d}{dx}(x^3) = 3x^2. \]Here
\[ \frac{d}{dx} \]acts on the function \(x^3\).
7. Comparing the notations
If
\[ y=f(x)=x^3, \]then all of the following express the same result:
8. “With respect to \(x\)”
The variable after the \(d\) in the denominator tells us which variable is treated as the independent variable.
For example,
\[ \frac{dy}{dx} \]means the rate of change of \(y\) with respect to \(x\).
But if a quantity \(s\) depends on time \(t\), we write
\[ \frac{ds}{dt}. \]This means the rate of change of \(s\) with respect to time.
9. The derivative at a particular point
The expression
\[ f'(x) \]gives the derivative as a function of \(x\).
To find the derivative at \(x=a\), write
\[ f'(a). \]In Leibniz notation, this may be written
The vertical bar means that the derivative is evaluated at \(x=a\).
10. Example
Suppose
\[ y=x^2. \]Then
\[ \frac{dy}{dx}=2x. \]At \(x=3\),
\[ \left. \frac{dy}{dx} \right|_{x=3} = 2(3) = 6. \]So the tangent to \(y=x^2\) has gradient \(6\) at \(x=3\).
11. Higher derivatives
A function can be differentiated again.
If
\[ y=f(x), \]the second derivative may be written
It describes how the first derivative itself changes with \(x\).
For example, if
\[ y=x^3, \]then
\[ \frac{dy}{dx}=3x^2 \]and
\[ \frac{d^2y}{dx^2}=6x. \]12. A useful physical interpretation
Suppose position is \(s(t)\).
Then
\[ \frac{ds}{dt} \]is velocity.
Differentiating again gives
\[ \frac{d^2s}{dt^2}, \]which is acceleration.
13. Keep the meanings separate
These three expressions are closely related, but they should not be confused:
Understanding this distinction prevents many common difficulties later in calculus and differential equations.