Introduction to Calculus
16C Rates of change
17A The derivative function
17B Derivatives at a given x-value
Today’s objectives:
(1) to understand an average and instantaneous rate of change
(2) to use limits to determine the gradient function or derivative at a
general point or a given x-value
RATES OF CHANGE:
A rate is a comparison between two quantities with different units. For example,
Speed is a commonly used rate.
average speed 
dis tan ce travelled
time taken
On a distance time graph, the average speed between any two points on the
curve is given by the slope of the curve.
If the graph is a straight line, the average speed is constant and given by the
gradient (slope) of the straight line. In this case, the speed of the car does not
vary over time and we can say that the instantaneous speed of the car is also
constant. In other words, the speed of the car is the same for the entire journey.
If the graph is a curve, the average speed between two points is the gradient
(slope) of the line joining the two points. In this case, the car’s instantaneous
speed will change during the journey. The average speed will not be the same as
the instantaneous speed of the car at every point. In fact, the instantaneous
speed is given by the gradient of the tangent to the curve at that time.
The Tangent to a Curve:
A chord or secant of a curve is a straight line segment which joins any two
points on the curve. The gradient of the chord [AB] measures the average rate of
change of the function for the given change in x-values.
A tangent is a straight line which touches a curve at a point. The gradient of the
tangent at point A measures the instantaneous rate of change of the function at
point A.
In the limit as B approaches A, the gradient of the chord [AB] will be the gradient
of the tangent at A.
The gradient of the tangent at x=a is defined as the gradient of the curve at
the point where x=a and is the instantaneous rate of change in f(x) with respect
to x at that point.
The Derivative Function
For a non-linear function with equation y  f (x) , the gradients of the tangents at
various points are different. Our task is to determine a gradient function so
that when we replace x by some value a then we will be able to find the gradient
of the tangent to y  f (x) at x=a.
Finding the gradient of a curve at any point can be done by finding the
gradient of the tangent line at that point. But how exactly do we do this?
Answer: By first finding the slope between two points on the curve and then
using limits, and our imagination to move those two points closer and closer
together.
Consider a function y  f (x) where A is (x, f (x)) and B is (x  h, f (x  h))
The chord [AB] has gradient =

f (x  h)  f (x)
xhx
f (x  h)  f (x)
h
If we now let B approach A, then the gradient of [AB] approaches the gradient of
the tangent at A
So the gradient of the tangent at the variable point (x, f (x)) is the limiting value
f (x  h)  f (x)
of
as h approaches 0 or
h
lim
h0
f (x  h)  f (x)
h
This formula gives the gradient of the tangent for any value of the variable
x. Since there is only one value of the gradient for each value of x, the
formula is actually a function.
This function is called the gradient function, derived function, derivative
function or more commonly, the DERIVATIVE.
The derivative of f(x) with respect to x has notation f (x) or
by:
dy
f (x  h)  f (x)
 f (x)  lim
dx
h
h0
To find the gradient function f (x) we need to evaluate the limit
f (x  h)  f (x)
. We call this the method of first principles.
lim
h
h0
Eg. Given f(x)=2x, find from first principles f (x)
dy
and is given
dx
Derivatives at a Given x-Value
Suppose we are given a function y=f(x) and are asked to find the derivative at a
point where x=a. This is actually the gradient (or slope) of the tangent to the
curve at x=a which we write as f (a)
We can find f (a) using first principles:
Since f (x)  lim
h0
f (a)  lim
h0
f (x  h)  f (x)
h
we can sub x=a and get
f (a  h)  f (a)
h
f (a)
A second method for finding
is to consider two points on the graph of
A(a, f (a))
B(x, f (x))
y=f(x): a fixed point
and a variable point
The gradient of chord [AB]=
f (x)  f (a)
xa
In the limit as B approaches A, x  a and the gradient of chord [AB]  the
gradient of the tangent at A
Thus
f (a)  lim
xa
the tangent at x=a
f (x)  f (a)
xa
is an alternative definition for the gradient of
Finally, you can find the derivative of a function at a given point using your TI
84.The easiest way to do this is to graph the function, then hit the following
keystrokes:
2nd TRACE  dy/dx
Then input the value of x at which you wish to find the derivative.