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Differentiation
Differentiation is the process of finding a derivative using the concept of a limit. This is explained here ...
Introduction
The definition of a derivative is the limit as an increment of the independent variable tends towards zero of the rate of change of the dependent variable with respect to the independent variable.
So, if the independent variable is and the dependent variable is
then the derivative of y
with respect to x is
Equation
(1)
Now, this looks quite scary, doesn’t it? But it all becomes easier when we take it easy and look at the component parts. Remember … break big problems down into smaller ones! Differentiation is the same as anything else in this way!
Well, take a look at the graph below:
In equation (1), the term is just a change in
and it is usually a small value. It would typically be the small change in
-value in moving from the point A on the curve
to the point B that is close to A. In
other words:
In the same way, is the (small) change in
moving from A to B, so
.
So, the value is just the rate of change of
with respect to
in the region of points A and B. It is also
the slope of the green line which passes through points A and B.
Now suppose the point B is gradually moved towards A, as in the figure below.
Note how the arrows alongside the axes show the movement in x
and y as B moves through C and D, towards A. In this case, I have used
letters as subscripts on the x- and y-values. A is the fixed point on the graph
of and the other points are meant to show a
progression of points getting closer and closer to A.
In this case, when we are considering the points A and B
together (I have labelled this
in the figure to show that the increment
refers to points A and B),
for points A and C, and
for points A and D. As the ‘moving’ points gets closer to point
A,
approaches zero (
). This
is what is meant by the concept
in Equation (1).
In other words, if we have a fixed point (point A) and let a
‘moving’ point approach point A, whilst keeping track of the slope of the line
passing through A and the moving point, the derivative of the function, ,
is the limit to which the slope tends.
This is a rather difficult concept to grasp, because you may
ask “Why not let the moving point become A itself?”. Well this cannot be done because then both and
would be zero and the ratio
would become
,
which is nonsensical!
The best way, I find, to follow this idea is by
visualisation. Imagine 2 points on the
curve and imagine there is a piece of taut elastic between them. This elastic represents the chord between A
and B in the last figure. Its slope is for the points A and B. Now keep A fixed and move B towards A
slightly. The elastic must remain taut
at all times. The chord will change as
the moving points travels towards A.
More importantly, its slope changes.
Now keeping A fixed, keep moving closer to A, and closer, … and
closer.
We’ll shortly come back to this idea …
Now imagine drawing a single straight line which passes through A and which is ‘in line’ with the curve at A. This line ‘skims’ point A. [An everyday example of this is how the road meets a bicycle wheel and how you would draw it in 2D cartoon form. You would draw a circle for the wheel and then draw a straight line under the wheel which touches the wheel only at one point (see figure below)]. Such a line is called a tangent line.
Now return to the ‘changing chord’ concept and, as the moving point approaches A, the chord approaches the tangent line. So the derivative at A is the gradient of the tangent line at A.
Well, that's the theory dealt with. Now you are ready to move on to the next section and put these ideas into practise with a specific example!
Note that there are a number of ways in which Equation (1) could be expressed. To see these, with an explanation of each, see my differentiation notation page.
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