Higher Derivatives

The Second Derivative

The derivative of the derivative is called the second derivative.  

There are two main ways of writing the second derivative.  They are

                                      d²y  
         f ''(x)         and                
                                        dx²

The main benefit of the first notation is that it is easy to write and understand, while the second is harder to understand but emphasizes which is the independent and which is the dependent variable.

Example:

Let 

        f(x)   =  3x⁴ - 2x³ + 5x² + x + 6

Find f ''(x)

Solution

To find the second derivative, we must first find the first derivative.  We have

        f '(x)  =  12x³ - 6x² + 10x + 1

Now take the derivative of the first derivative to find the second derivative

        f ''(x)  =  (12x³ - 6x² + 10x + 1)'  =  36x² - 12x + 10

Exercises   

Find f ''(x) if

A)                          3
            f(x)  =                     
                          2x - 1

B)         f(x)  =  (x² - 4)⁵

Higher Dervatives

Just as we can take a second derivative, we can take a third, fourth, fifth, etc. derivative.  The notation is similar.  The third derivative is written as

                                       d³y  
         f '''(x)         and                
                                         dx³

For the fourth derivative and on, we do not use the cumbersome primes.  Instead, use a superscript embraced by parentheses.  For example the seventh derivative is written as

                                       d⁷y  
         f ⁽⁷⁾(x)         and                
                                         dx⁷

Example

Find f '''(2) when 

        f(x)  = x⁴ - 3x² + 9

Solution

To find the third derivative, we find the derivative, then the derivative of the derivative, and finally the derivative of the derivative of the derivative.  We have

        f '(x)  =  4x³ - 6x        and        f ''(x)  =  12x² - 6

Taking one more derivative gives

        f '''(x)  =  24x

Now plug in x  =  2 to get

        f '''(2)  =  24(2)  =  48

Acceleration

Recall that if s(t) is the position function, then v(t) = s'(t) is the velocity function.  We define the acceleration function as

        a(t) = v'(t) = s''(t).

Example:

A baseball is hit into the air and has position function

        s(t) = -16t² + 25t + 4

Find the velocity and acceleration

Solution

We have

        v(t) = s'(t) = -32t + 25

        a(t) = s''(t) = v'(t) = -32

Exercise

An raindrop falling from a cloud 1000 feet above the ground has approximate position function

        s(t)  =  1000 - 16 t² + 0.7 t³

where t is measured in seconds.

A.  Use your calculator to determine when the raindrop will hit the ground.   

B.  How fast is it going when it hits the ground?       

C.  What is it's acceleration when it hits the ground?       

(Hold your mouse over the yellow rectangle for the solution)

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