Section 12.2 Question 1

How is the second derivative calculated?

The second derivative is the derivative of the derivative. To help identify this process, the first time we take a derivative of a function we call it the first derivative. The first derivative, as we have seen earlier, is symbolized in several different ways. If we take the first derivative of a function y = f (x), the first derivative is written as12_2_1_1

Several notations for the second derivative are used.

Any of the following notations may be used to write the second derivative of a function :12_2_1_2

We can take the derivative of the first derivative by applying the rules for derivatives to the first derivative. If f (x) = 5x4 – 7x2 +2x – 1, then we can apply the Sum / Difference, Product with a Constant, and  Power Rules for Derivatives to yield the first derivative12_2_1_3

If we use these rules again on the first derivative, we get the second derivative,12_2_1_4

The process for finding the derivative is the same as we have used in Chapter 11. The only difference is the starting point. When finding the first derivative of a function, we take the derivative of the function. For the second derivative, we take the derivative of the first derivative.

Example 1      Calculate the Second Derivative

Let  f (x) = x3 -4x2 + 6x +12. Find the second derivative ″(x).

Solution The first derivative of  is found by applying rules for derivatives,12_2_1_5

To find ″(x), take the derivative of f ′(x) = 3x2 – 8x +6:

12_2_1_6

The second derivative is  f ″(x) = 6x – 8.


Example 2     Calculate the Second Derivative

Let12_2_1_7. Find the second derivative 12_2_1_8.

Solution The first derivative is found with the Product Rule for Derivatives using the factors

12_2_1_9

The first derivative is

12_2_1_10

Apply the product rule again to find the second derivative with the factors

12_2_1_11

The second derivative is

12_2_1_12

The second derivative is 12_2_1_13.


Example 3      Calculate the Second Derivative

Let 12_2_1_14. Find the second derivative g″(x).

Solution Use the Quotient Rule for Derivatives to find the first derivative with

12_2_1_15

The first derivative is

12_2_1_16

To compute the second derivative, take the derivative of the first derivative with the Quotient Rule for Derivatives,

12_2_1_17

The second derivative is

12_2_1_18

The second derivative expression can be simplified further by factoring the numerator:

12_2_1_19

The second derivative is 12_2_1_20.


Goto How are other higher derivatives calculated?

Goto Beginning of Section 12.2

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