finding the derivative the limit process. what is the derivative of something? the derivative of a...
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Finding the Derivative
The Limit Process
What is the derivative of something?
The derivative of a function f(x) is, mathematically speaking, the slope of the line tangent to f(x) at any point x. It is also called “the instantaneous rate of change” of a function.It can be equated with many real world applications such as;Velocity which is speed such as miles per hourIn business, the derivative is called marginal, such as the marginal Cost function, etc.
We call a line which intersects a graph in two points a secant line. It is easy to find the slope of that line. But it takes limits in order to find the slope of a tangent line which touches the graph at only one point.
The green line is a secant line because it crosses the blue graph more than once. In particular we focus on the two points illustrated. For animation you need to be connected to the internet.
A tangent to a graph.
To find the slope of the tangent line to a graph f(x) we use the following formula:
h
xfhxfh
)(lim0
First let’s see how this formula equates with the slope of a tangent line.
Notation: The derivative of f(x) is denoted by the following forms:
dx
dy
xf or
h
xfhxfh
)(lim0
(x,f(x))
(x+h,f(x+h))
x x+h
The slope of the secant line would be
h
xfhxf
xhx
xfhxf )()(
)(
)()(
By decreasing h a little each time we get closer and closer to the slope of the tangent line. By using limits and letting h approach 0 we get the actual slope of the tangent line.
The difference quotient
measures the average rate of change of y with respect to x over the interval [x,x+h]
h
xfhxf )(
In a problem pay attention to that word average. If it is there then you do not use limits.
Example 1:
Let xxxf 6)( 2
Find the derivative f’ of f.
Substitution of numerator.
Multiplying out
Combining like terms
Factor out h
Cancel
Let h = 0. Done.
6262lim
)62(lim
)62(lim
62lim
6662lim
6662lim
6)(6)(lim
)()(lim
6)(
0
0
0
2
0
222
0
222
0
22
0
0
2
xhxh
hxhh
hxhh
hhxh
h
xxhxhxhx
h
xxhxhxhx
h
xxhxhx
h
xfhxf
xxxf
h
h
h
h
h
h
h
h
Find the derivative f’ of f.
Examples
1. f(x) = x3
2. f(x) = x2 + 2
Apply the definition of the derivative