nth Derivative of the Radical of a Linear Factor
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Find the
derivative of
.
How do we proceed?
We determine the first, second, third, and fourth derivatives. Then we can usually see the pattern needed for the
derivative.
How do we take the derivative of a radical?
We first convert it to an equivalent expression in terms of a power.
Do that.
What rules of differentiation do we use here?
The Power Rule and the Chain Rule.
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
Determine
.
Simplify this expression.
What pattern does the exponent follow?
It is reduced by 1 in each stage. Because it starts at
, the successive values are in the form of an odd negative number over 2.
What is the pattern of the overall algebraic sign?
It alternates between positive and negative.
Do we take the derivative of
at each stage?
Yes.
What is the consequent pattern?
We get another power of 3 as a factor each time.
Does the previous power become a factor each time?
Yes, because of the Power Rule.
To confirm this pattern, determine
.
Simplify this expression.
.
The alternating sign and other patterns are there.
How can we express this for the
term?
We can use a factor
to produce a positive sign when
n
is an odd number, as 5 is here.
Will that work when
n
is an even number?
Yes, that is the case where we need an odd power to produce a factor of – 1.
How can we handle the power on the 3 for the
term?
We can use
.
What can we use for the power of 2 in the
term?
We can use
.
How can we handle the exponent on
?
It becomes
.
From above, we see that we need a series of factors ( 5*7*9*….*(2
n
– 3) ).
Combine this analysis to determine
.
Is this the only form of the result?
No. Notice that
could also be used to handle the algebraic sign. We also might use
and include the other 3 in the string of factors:
.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index