The derivative of a function is defined as the instantaneous rate of change of the function at a certain point. For a line, this is just the slope. For more complex curves, we can find the rate of change between two points on the curve easily since we can draw a line through them.
In the image above, the average rate of change between the two points is the slope of the line that goes through them: .
We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the limit as goes to 0 will give us the derivative of the function at :
If this limit exists, it is the derivative of at . If it does not exist, we say that is not differentiable at . This limit is called Fermat's difference quotient.
We can apply the Fermat's difference quotient to a polynomial of the form in order to find its derivative. If we imagine the secant line intersecting a curve at the points and . Then we can change this to the tangent by setting on top of . Let us call the horizontal or vertical distance as .
After canceling like terms we should have all terms contain an . We can then cancel out the and set . Our end result is the first-derivative.
The first derivative is denoted as .
This would be some tedious work so instead there is a much nicer way to find the derivative.
Let . Let
1. Find .
Any function like this is:
2. Find .
Breaking apart on what we used above.
Let . Find .
If the function is a constant then its derivative will always be .
Notation: denotes the first derivative for . The symbol for the second derivative is just . For the third derivative it is just . Derivatives are also written as . Or if for the nth derivative they are written as .
Maximum and Minimum: We can use the first derivative to determine the maximum and the minimum points of a graph.
If . Then the maximum and the minimum occur when:
, or . We can plug each back in to the original if it was given, and the one with the higher y-coordinate is the maximum, while the smaller y-coordinate gives the minimum.
Below are problems for Part I. In Part II(see link below) we will begin to actually "start" the calculus with this.
: Find the first derivative of , where .
: Find the equation of the line tangent to the function at the point .
We will take the first derivative to determine the slope of the tangent line.
. If this is the slope of the tangent point then we can just plug into the coordinate to find the actual slope.
. The slope of the line is .
Let the equation be:
Plugging in gives and so .
Thus, the equation of the line is . Alternatively, one could use point-slope form for the line; after determining that the slope is , as above, this allows one to immediately write down the equation of the line.
(Notice that it is implicit in the question that the point lies on the graph of ; it's easy to check that this is actually the case.)
: Find the nth derivative of
The nth derivative of is .