We have just looked at describing and estimating the instantaneous rate of change of a function. Describing the rate of change of a function is the key concept in differential calculus and as such we have language and notation around the processes involved.
Modern calculus was first developed in the late 17th century independently by Isaac Newton and Gottfried Wilhelm Leibniz and two distinct notations are still in common use today.
The function which gives the gradient of the tangent to the graph of y=f(x) at any point is called the derivative of f(x). The derivative is sometimes also referred to as the gradient function of f(x). The process of finding the the gradient function from the function f(x) is called differentiation and not surprisingly is the focus of differential calculus. As the derivative gives us the gradient of the tangent to a function we can use it to calculate instantaneous rates of change.
These phrases all require the same process to be carried out:
- Differentiate...
- Find the derivative of ...
- Find the gradient function for ....
Leibniz's notation
The first notation we shall look at was first used by Leibniz and remains popular to this day.
The gradient between two points can be represented by the notation m=\frac{\Delta y}{\Delta x}. \Delta is the Greek upper case letter d (delta) and in mathematics represent "the change in". So, \frac{\Delta y}{\Delta x} reads "the change in y over the change in x", hence why this denotes the gradient between two points. Using the lower case delta the expression \frac{\delta y}{\delta x} is the gradient between two points which are very close, that is "a small change in x and the corresponding small change in y". Taking the small change in to the limit, we obtain the gradient at a point - the instantaneous rate of change which is given the notation \frac{dy}{dx}.
Thus, for a function y=f(x), we denote the derivative as \frac{dy}{dx}, pronounced dee y dee x. This uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively. So the symbol is representing change in y over change in x - which makes sense since the function represents the gradient. We can also read the symbol \frac{dy}{dx} as "the derivative of y with respect to x".
If functions have different variables, for example a function describing population over time, such as P=200\times1.08^t, then the derivative can be written as \frac{dP}{dt}. This is the derivative of P with respect to t or "dee P dee t" and the derivative would describe the rate of change of the population.
Lastly, we can use this notation as an operator, that is \frac{d}{dx}\left(...\right), means to differentiate what follows in the brackets with respect to x.
Example 1
For y = x^2, the gradient function is \frac{dy}{dx}=2x. This means the gradient of the tangent at any point is double the x-coordinate. For example, the gradient of the tangent to y=x^2 at x=3 is:
| At x=3,\frac{dy}{dx} | = | 2\times3 |
| = | 6 |
Rather than stating "At x=3" shorthand for evaluating a derivative at a point for this notation is:
Lagrange's notation
Lagrange's notation is one of the most commonly used in calculus. Lagrange was an Italian mathematician and astronomer who made popular this notation.
For a function f(x), we denote the derivative as f'(x), pronounced f primed of x or f dashed of x. Again we can functions and their derivatives using different variables such as the function d(t)=4.9t^2, giving the distance an object has fallen after t seconds and then its derivative d'(t)=9.8t, which will give the rate of change of distance with respect to time of the object falling - its speed.
Example 2
For f(x) = 4x^3, the gradient function is f'(x)=12x^2. This means the gradient of the tangent to f(x)=4x^3 at x=2 is:
| f'\left(2\right) | = | 12\times2^2 |
| = | 48 |
Notice: With Lagrange's notation evaluating the derivative at a point is just like function notation. f'(a), means evaluate the derivative of f with respect to x, at the point where x is equal to a.
Further investigations
This is a very brief look at the two main notations used in calculus, further things to investigate are:
- Are there other notations in use? Did Newton have a different notation and is it still used?
- At the time of its inception it was controversial to whether Newton or Leibniz had first invented calculus. Is there any evidence to suggest the idea was plagiarised?
- What is the notation for taking a second or third derivative in the different notations? For example taking the derivative of a displacement function will give a velocity function and then the derivative of a velocity function will give the acceleration function. The derivative of acceleration also has a name - can you find it?
- Do all functions have derivatives?