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| Isaac Newton |
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| Gottfried Leibniz |
Both men claimed to have been the first to make the discovery and a bitter dispute developed. As a result, we use a mix of notation when differentiating!
Use the search boxes at the side of the page to find answers to the following questions:
1: Why was calculus developed? What problem was it initially designed to resolve?
2: Can you find any examples of Differentiation in action in the world today?
Simply put, differentiation helps us to find the rate of change of one quantity with respect to another.
A very simple example: if a tree is 1.5 meters in year 1 and after 3 years, it has reached a height of 3 meters, then the rate of change in height, with respect to time in years, is ½ meter per year.
This however, is a very straightforward example, as what I have described is a linear equation (if you're not sure what a linear equation is, use those search engines!). What we are talking about when we say "rate of change" is actually the slope, or gradient, of the equation.
Remember:
When the rate of change is not constant (non-linear), then things become a little trickier. For example, think about the rate of growth of a child in the first months of life. The length (height) of a particular gitl in the the first 36 months of life is represented by the following growth chart.
As you can see, the graph is a curve and the slope of a curve is not constant. So what does it mean to talk about the rate of change of a curve? Well the slope, or steepness, of a curve varies, so that means that the rate of change of the curve also varies. In the example above, is the slope positive, negative or both? At what stage does the child grow the fastest?
Differentiation is the method that we use to find the equation of the slope of non-linear functions. When the slope is not constant, its equation will be a function that will return different values at different points along the graph. What we will be calculating is the equation of the slope of the tangent at any point along the curve.
Let's investigate the a curve that could be generated by throwing an object straight up into the air. As it goes up, the object slows down, then reverses direction and falls back to earth. The graph would look like this:
The slope or gradient (steepness) of the curve changes throughout the motion, representing the velocity, or speed of the object. Initially, it has a steep, positive slope. As the object slows down, the slope decreases until, when the object reaches its highest point, the slope is 0. Then, as the object starts to fall, the slope becomes negative and steeper as gravity pulls the object back to earth.
However, given that we need two points on a line in order to calculate a slope, how do we to calculate the slope of a tangent, given only one point on the curve? Let zoom in to the graph of the object in motion and pick a random point on the curve, call it A, with coordinates (x, f(x)), and another point on the curve, B, (x + h, f(x + h)) which is close to A. The distance between these two points is dependent on the value of h, which is the difference between their x coordinates.
Change the value of h using the slider.
What happens as h approaches 0?
What happens when h reaches 0?
As h approaches 0, the slope of the line AB approaches the slope of the tangent to the curve at the point A. However, as division by 0 is undefined (and therefore not allowed), we look at the value of the slope as h gets very small. This is known as the limit of the function. The limit allows us to make statemetns about the formula that we would not otherwise be able to make. While we cannot say anything about the slope when h euals zero, we are allowed to talk about the value that the slope approaches as h approaches zero. We use the usual slope formula, but couch it inside a limit as h approaches 0. This limiting value is called the first derivative of y with respect to x and is denoted by f '(x), or by dy/dx
Using the slope formula, we can say that the slope of the line AB is described by:
(x + h) - x can of course be simplified to h, giving us
Right, now we are ready to tackle a couple of questions! Let's see if we can differentiate the function
f(x) = x2 from first principles. The way to approach this kind of question, is to do it in a step by step fashion.
1) we know f(x) = x2
2) calculate f(x + h).
f(x + h) = (x + h)2
= x2 + 2xh + h2
3) calculate f(x + h) - f(x) to get the top line, or numerator, of the formula
f(x + h) - f(x) = x2 + 2xh + h2 -x2
= 2xh + h2
4) divide f(x + h) - f(x) by h to complete the fraction.
5) The final step in this process is to calculate the limit as h approaches 0.
As h → 0, the expression 2x + h → 2x.
Therefore, if f (x) = x2 then the derivative of f(x) is 2x.
f '(x) = 2x
The following graphic traces the equation of the derivative of the f(x) = x2. Drag the slider a to move the cursor around the curve and S will draw the equation of the derivative. You can check that the derivative is correct by entering g(x) = 2x into the input bar. If the line that is generated is the same as the trace generated by S, then our calculation is correct!
You can change the function being displayed in the above graphic, by typing f(x) = new function into the input bar, where new function is whatever function you wish to graph. Use Ctrl+f to clear the current trace.
Try this for example, with
(a) f(x) = x^3 (which represents x3, as the ^ notation means 'to the power of'
(b) f(x) = 1 - x^2
(c) f(x) = 1/x
(d) f(x) = 2x^4 + 3x^2 -1
Try to come up with a conjecture about the relationship between the graphs of the functions and those of their derivatives.
Now, attempt to differentiate the following functions from first principles
(a) Differentiate f (x) = x3
show answer
(b) Differentiate f (x) = 2x3 + 3x + 2 from first principles.
show answer
