Section 12.2 Question 2

How are other higher derivatives calculated?

The second derivative is an example of a higher derivative. By taking the derivative of the second derivative, we obtain another higher derivative, the third derivative. The notation for the third derivative follows the pattern established by the second derivative.

Any of the following notations may be used to write the third derivative of a function y = (x):
12_2_2_1

By taking the derivative of another derivative, we may calculate other higher derivatives. These higher derivatives may be written in a similar manner. For derivatives higher than the third derivative, we write the nth derivative as f (n)(x).

For the function (x) = 5x4 – 7x2 + 2x – 1, the first and second derivatives were calculated earlier as

12_2_2_2

The derivative of the second derivative, the third derivative, is calculated by taking the derivative of the second derivative. We can continue to take the derivative of the derivative to obtain higher derivatives:

12_2_2_3

Since the derivative of a constant is zero, the fifth derivative and those higher are equal to zero.

Example 4      Calculate the Fourth Derivative

Let

12_2_2_4

Find the fourth derivative h(4) (x).

Solution The first derivative is

12_2_2_5

We can follow the same strategy to get the second derivative,

12_2_2_6

the third derivative,

12_2_2_7

and the fourth derivative,

12_2_2_8


Example 5     Calculate the Third Derivative

Let12_2_2_9. Find the third derivative 12_2_2_10.

Solution To find the first derivative, use the Product Rule for Derivatives with the factors and corresponding derivatives,

12_2_2_11

The product rule leads to12_2_2_12

The second derivative is simply the derivative of the first derivative or

12_2_2_13

Rewrite 12_2_2_14 as -1 to make the next derivative easy to do. The third derivative is

12_2_2_15

The third derivative is rewritten with a positive exponent as 12_2_2_16.


Goto What does the second derivative of a function tell you about a function?

Goto Beginning of Section 12.2

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