nth Derivative of the Negative 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.
How do we take the derivative of a power in the denominator?
We first convert it to an equivalent expression in terms of a positive power.
Do that.
What rules of differentiation do we use here?
The Power Rule and the 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 simplify product 3*4*5*6 with 6 factorial, 6!
Yes.
How?
Since 6! = 6*5*4*3*2*1, we can write
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 ( – 1) as a factor each time.
What is the pattern of the overall algebraic sign?
It stays positive because of the negative power becoming a factor. This gives us two negative factors each time we take a derivative.
To confirm this pattern, determine
.
Simplify this expression.
We get
Express
We get
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General Contents
Detailed Contents
Index