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First and Second Derivatives

by mmajames
Tags: derivatives
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mmajames
#1
Mar6-12, 09:50 PM
P: 12
1. The problem statement, all variables and given/known data
Just trying to find the first and second derivatives.

X^2/(X^2-16)

1+X/1-X

X^3(X-2)^2

2. Relevant equations
Quotient Rule/Power Rule/Chain Rule


3. The attempt at a solution
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80past2
#2
Mar6-12, 09:51 PM
P: 40
For the first one, what is the derivative of f(x)/g(x)? In general, not for this specific function.
mmajames
#3
Mar6-12, 09:54 PM
P: 12
(g(x)*f'(x)-f(x)*g'(x))/g(x)^2

80past2
#4
Mar6-12, 09:56 PM
P: 40
First and Second Derivatives

okay, so if f(x) = x^2 and g(x) = x^-2, what is the derivative of f(x)/g(x)?
mmajames
#5
Mar6-12, 10:06 PM
P: 12
f(x) = x^2 and g(x) = x^-2
x^2/x^-2
((x^-2)(2x)-(-2x)(x^2))/ (x^2)^2
80past2
#6
Mar6-12, 10:30 PM
P: 40
I mistyped (that g(x) should have been g(x)=x^2 - 16 [not sure how I messed that one up]), but you seem to know what you're doing. What specific question do you have?
mmajames
#7
Mar6-12, 10:35 PM
P: 12
I'm mostly having trouble with the second derivatives.
80past2
#8
Mar6-12, 11:10 PM
P: 40
What do you have so far? Where are you getting stuck?
mmajames
#9
Mar8-12, 07:17 PM
P: 12
Well here's what I've got, I think they're right but I'm not sure.

f(x)=X^2/(x^2-16)
(X^2-16)(2X)-(X^2)(2X)/(X^2-16)^2
f'(x)=-32X^2/(X^2-16)^2
(X^4-32X^2+256)(-64X)-(-32X^2)(4X^3-64X)/(X^4-32X^2+256)^2
-64x^5+2048X^3-16384X+128X^5-2048X^3
f''(x)=64X^5-16384X/(X^4-32X^2+256)^2

f(x)=1+x/1-X
(1-X)(1)-(1+X)(-1)/(1-X)^2
f'(x)=2-2X/(1-X)^2
(X^2-2X+1)(-2)-(2X-2)(2-2X)/(X^2-2X+1)^2
f''(x)=2X^2+4X+2/(X^2-2X+1)^2

f(x)=X^3(X-2)^2

(X^3)(2X-4)+(3X^2)(X-2)^2
2X^4-4X^3+3X^4-12X^3+12X^2
f'(x)=5X^4-16X^3+12X^2
5X^4-16^3+12X^2
f''(x)=20X^3-48X^2+24X
mmajames
#10
Mar8-12, 09:09 PM
P: 12
Am I doing these correctly?


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