3.8 Higher Derivatives

*The second derivative is the derivative of (x) [the derivative of f]. We write it as

And the process continues with higher derivatives….

Example 1

Find the fourth derivative of y

y=x³ –4x²-6x+7

= 3x²-8x-6

= 6x-8

=6

y⁴= 0

Example 2

Find if x⁶+y⁶=36

use implicit differentiation

6x⁵+6y⁵y¹= 0

Solving for gives us:

=

To find differentiate for using quotient rule. Remember y is a function of x.

u= v=

=

Now substitute equation 1 into the expression

Get a common denominator to the numerator.

Simplify.

Factor.

Substitute 36 in for

= ß Final Answer

Example 3

Find D²⁷cosx

The first derivatives of cosx are:

Dcosx= -sinx

D²cosx=-cosx

D³cosx=sinx

D⁴cosx=cosx

D⁵cosx=-sinx

We see that the successive derivatives occur in a cycle of length 4 and in particular Dⁿcosx=cosx whenever n is a multiple of 4. Therefore,

D²⁴cosx=cosx

And differentiating 3 more times we have

D²⁷cosx=sinx ß Final Answer