nth Derivative of the nth Power of a Linear Factor
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Find the
derivative of
.
How do we interpret the symbols
? Is multiplication involved?
No, this group of symbols describes a function named
f
which depends on the variable
x
.
We frequently use the variable
y
for our dependent variable. What is the dependent variable here?
The function
is the dependent variable here.
How do we proceed?
We determine the first, second, third, and fourth derivatives. Then we can usually see the pattern needed for the
derivative.
What rules of differentiation do we use here?
The Power Rule and Chain Rule.
Determine
.
Why do we have that last factor?
It indicates the application of the Chain Rule.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Lets analyze what is happening here. Does the previous power become a factor each time?
Yes, because of the Power Rule.
Can we do something like this at each stage?
Yes.
What pattern does the exponent follow?
It is reduced by 1 in each stage.
Do we take the derivative of
at each stage?
Yes.
What is the consequent pattern?
We get another power of 2 as a factor each time.
Use these ideas to find the
j
-th derivative, where
j
is less than
n
.
We get
Simplify this expression.
We get
What happens to the exponent when
?
The exponent becomes 0 and we have
.
What is the value of (anything)0?
(anything)0 = 1
What will the last factor,
, become when
?
It will become equal to 1.
Use these special results to determine the
derivative.
Can we the simplify product
?
Yes.
How?
It is
n
!
Is this result a constant?
Yes, there is no longer any dependence on the factor (2
x
+ 1).
Determine the next derivative,
.
Since the derivative of a constant is 0, we get 0 for all derivatives beyond the
in this special case.
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General Contents
Detailed Contents
Index