Find the n'th derivative of a function

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Find the n'th derivative of a function

The question asks:
Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

the function is $f(x)\, =\, x^{(n)}$
So I need to find the function $f^{(n)}(x)$

So if I start taking the derivative I get:
$f'(x)\,=\,nx^{n-1}$
$f''(x)\,=\,(n-1)nx^{n-2}$
$f'''(x)\,=\,n(n-1)(n-2)x^{n-3}$

so on and so forth. I see the pattern, but I really don't know the notation to write it out as one function.
~ Jared Beach ~

jaredbeach
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Re: Find the n'th derivative of a function

jaredbeach wrote:The question asks:
Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

the function is $f(x)\, =\, x^{(n)}$
So I need to find the function $f^{(n)}(x)$

So if I start taking the derivative I get:
$f'(x)\,=\,nx^{n-1}$
$f''(x)\,=\,(n-1)nx^{n-2}$
$f'''(x)\,=\,n(n-1)(n-2)x^{n-3}$

so on and so forth. I see the pattern, but I really don't know the notation to write it out as one function.

$\text{Use the factorial notation where : } n!=n(n-1)(n-2)...\text{3x2x1.}$

zaidalyafey
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Re: Find the n'th derivative of a function

Yes, the factorial is one way to go. Another is the permutation function:

$n(n-1)(n-2)\cdots(n-k)=\text{nPr}(n,k)$
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
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Re: Find the n'th derivative of a function

zaidalyafey wrote:
jaredbeach wrote:The question asks:
Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

the function is $f(x)\, =\, x^{(n)}$
So I need to find the function $f^{(n)}(x)$

So if I start taking the derivative I get:
$f'(x)\,=\,nx^{n-1}$
$f''(x)\,=\,(n-1)nx^{n-2}$
$f'''(x)\,=\,n(n-1)(n-2)x^{n-3}$

so on and so forth. I see the pattern, but I really don't know the notation to write it out as one function.

$\text{Use the factorial notation where : } n!=n(n-1)(n-2)...\text{3x2x1.}$

So is the answer$n!x^{n-1}$?
~ Jared Beach ~

jaredbeach
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Posts: 93
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Location: Alabama

Re: Find the n'th derivative of a function

Where $\rightarrow$ means to take the next derivative. This notation is NOT standard and is used for illustration only:

$f(x)\rightarrow 1 =1!$
$f(x^2) \rightarrow 2x \rightarrow 2 = 2!$
$f(x^3) \rightarrow 3x^2 \rightarrow 6x \rightarrow 6 = 3!$

jks
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Re: Find the n'th derivative of a function

Oh! so it's just $n!$?
~ Jared Beach ~

jaredbeach
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Posts: 93
Joined: Wed Aug 17, 2011 10:05 am
Location: Alabama

Re: Find the n'th derivative of a function

jaredbeach wrote:
zaidalyafey wrote:
jaredbeach wrote:The question asks:
Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

the function is $f(x)\, =\, x^{(n)}$
So I need to find the function $f^{(n)}(x)$

So if I start taking the derivative I get:
$f'(x)\,=\,nx^{n-1}$
$f''(x)\,=\,(n-1)nx^{n-2}$
$f'''(x)\,=\,n(n-1)(n-2)x^{n-3}$

so on and so forth. I see the pattern, but I really don't know the notation to write it out as one function.

$\text{Use the factorial notation where : } n!=n(n-1)(n-2)...\text{3x2x1.}$

So is the answer$n!x^{n-1}$?

I suspect what you meant is:

$f^{\small{(n)}}(x)=n!x^{n-n}=n!$
Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

MarkFL
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