Just as there is a notation for the first derivative so there is a similar notation for higher derivatives. Consider the function, . We know that the first derivative is or which is the instruction to differentiate the function . The second derivative is calculated by differentiating the first derivative, that is
So, using a fairly obvious adaptation of our derivative notation, the second derivative is denoted by and is read as ‘dee two by dee squared’. This is often written more concisely as .
In similar manner, the third derivative is denoted by or and so on. So, referring to Example 6 we could have written
If then its first, second and third derivatives are denoted by:
In most examples we use to denote the independent variable and the dependent variable. However, in many applications, time is the independent variable. In this case a special notation is used for derivatives. Derivatives with respect to are often indicated using a dot notation, so can be written as , pronounced ‘ dot’. Similarly, a second derivative with respect to can be written as , pronounced ‘ double dot’.
Calculate and given .
First find :
Now obtain the second derivative:
Finally, obtain the third derivative:
Note that in the last Task we could have used the dot notation and written , and
We may need to evaluate higher derivatives at specific points. We use an obvious notation.
The second derivative of , evaluated at say, , is written as , or more simply as . The third derivative evaluated at is written as or .
First find and :
Now substitute in to obtain :
. Remember, in the ‘ ’ is radian.
Now find :
Finally, find :
is defined by:
- Find where is given in Exercise 1.
is given by:
- Calculate for the functions given in Exercise 3.