In this tutorial
we shall discuss the concept of differentiation
or differential calculus. Differentiation can be
defined as the process of finding the derivative
of a function, which measures both the slope and
the instantaneous rate of change of the function
at a given point. The derivative of a function is
itself a function. Recall that a function is a set
of ordered pairs with the property that any x value
will uniquely determines a y value. This means that
in a function, for each value of x, there will be
a unique value of y. This suggests that when there
is a change in variable x (independent variable),
there will be a change in y (dependent variable).
The rate at
which y (the dependent variable) changes in relation
to the changes in x (the independent variable) is
Differentiation. In otherwords, the number of times
that y will increase or decrease as a result of
a change in x is what differentiation is all about.
That "number of times" is the derivative
of the function.

**Derivative
Notations**

The derivative of a function can be written in many
ways.

Given a function as y = ƒ(x), the derivative
of the function can be written as:

So, given the function
y = 2x + 3, the derivative of the function can be
expressed as:

Also, if the function
is given as s = t + 1, the derivation can be written
as:

Differential Coefficient

The differential coefficient of a function
is the slope or gradient of the function at a particular
point. Given a function, which defines the relationship
between two variables; y - dependant variable and
x - independent variable. A small change in the
value of x will cause a proportional change in y.
The rate at which y changes in respect to the change
in x is called the differential coefficient or slope
or gradient of the function. It is usually represented
as .
The small change in x is denoted by
while the small change in y is denoted by .
In computing
the differential coefficient of a function, we shall
add the small change in x i.e
to every x in the function and the resultant small
change in y i.e
to the y on the other side of the equation, and
then solve for .**
**Lets take some examples:

**Question**:
Given that y = 2x.
Find the differential coefficient.

**Answer**:
The given function is y = 2x. We shall add the change
in the variables to the respectives in the equation.
So, we will have:

Note that the differential
coefficient of the function, which has been computed
to be 2, is also the slope of the function.

You are expected to memorise the procedure that
was followed above. lets examine more examples in
the next page