Differentiation

 

The Basics

Differentiation is all about rates of change.

Keep in mind what the notation means: the change in y with respect to x.

In other words, for each unit that x changes, how much does y change?

 All the basic formulas are included in the formula book. These include:

 

 

 Applications for Graphs

When dealing with the graphs of functions, we use the 1^st derivative (e.g.) to find the gradient.

 When the gradient at a particular point is 0, we have a turning point. The 2^nd derivative is then used to

 investigate the nature of the turning point. These fall into 3 different categories:

 

MINIMA:

At the turning point, and

Note: the 2^nd derivative is +ve because the gradient increases as x increases.

 

MAXIMA:
At the turning point,and
Note: the 2^nd derivative is -ve because the gradient decreases as x increases.

 

HORIZONTAL POINT OF INFLECTION:

At the turning point, and
Note: the 2^nd derivative is 0 because at this point, the gradient is neither increasing nor decreasing.

   

Special Cases

 A stationary point that has the properties andcould, in fact, be any of these 3.
If it is a point of inflection, it will have the additional property that .
If you get a question where the 1^st,2^nd and 3^rd derivatives are all equal to 0, the point will be a MAX or MIN.

Simply find the sign of the gradient slightly to each side of the point. This will indicate which type it is.

 

 You can also get non-horizontal points of inflection, such as the tan function.

These have the property that, but , i.e. it's not flat:

 

 

 Differentiating Products, Quotients and Composite Functions

These rules are again in the formula book

The product rule is:

E.g. Differentiate

   

 The Quotient rule is:

E.g. Differentiate

[Note that]

 The Composite Rule is about functions of other functions.

For instance in the function , you're first multiplying x by 3, then finding the sin value. Use the rule, often called the chain rule, from the formula book:

  In our example,

 

Parametric Differentiation

We use the fact that and use the composite rule.

 The x and y values are differentiated with respect to the common parameter,

E.g. if then is the common parameter

 

 There are a couple of common parametric equations worth knowing:

 Where a and b are constants, is a parabola

 Is a circle with centre (a , b) and radius r