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e LEARNING - MATHEMATICS:

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Page 1

TOPIC: CALCULUS: DIFFERENTIATION

DIFFERENTIATION DEFINITION

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

 

 

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