The same rules apply for algebraic fractions as for normal fractions When adding or subtracting, make a common denominator. When multiplying fractions, cross cancel first if possible. Multiply the denominators together to make a new denominator. Multiply the numerators together to make a new numerator. Cancel if possible. When dividing fractions, invert the dividing fraction […]

## Exponential Functions

Exponential growth means growth without limit. The rate of growth of a quantity is directly proportional to the quantity itself and this leads to equations of the formwhereand are constants,represents the quantity and represents the time. In the long term of course, nothing ever grows without limit, a lesson bankers are learning and exponential growth […]

## Expressing Functions of the Formin the Form

Linear combinations of trigonometric formulae are very important: in fact any continuous function can be expressed as a sum of sin and cosine terms under certain conditions. Any function of the form can be expressed in the formor C can be found simply in any case:but forthe re are 4 possibilities. If we are expressing […]

## Implicit Differentiation

If we haveas a function ofit is quite easy to findWe often need to find whenis a function ofor there are several occurrence of bothandIn these cases we need to differentiate implicitly. We shall start with a simple case. Differentiate We can differentiate both sides with respect toobtaining 1 on the left hand side but […]

## Differentiation – The Chain Rule

If we have to differentiate a function which consists of one operation carried out after another we have to use the chain rule. Several examples are shown below together with the constituent functions which we call u and v. Function We differentiateandand substitute them into The Chain Rule: Example: Differentiate Now just multiply the differentiated […]

## Differentiation – The Product Rule

You may know hoe to differentiate a simple function such asorGenerally functions are built out of these simple functions to make more complicated functions and we must learn to differentiate these more complicated functions too. The simplest way two functions can be combined to make a more complicated function is to multiply them. Then they […]

## Partial Fractions

We may have to express expressions such as in the form Notice first that factorises into We can write, Ignore the first term here and multiply throughout by We get after cancellation. The procedure now is to eliminate the A term by putting x=2 Eliminate the B term by putting x=-4 Then If we have […]

## Absolute Value Graphs

Taking the absolute value of a number is equivalent to ignoring any minus sign, We are only interested in the magnitude of a number. For example, and We sometimes have to sketch graphs of the form The graph of is a straight line. When , . If , for example if , then which is […]