First and Second Derivatives

mmajames

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

80past2

For the first one, what is the derivative of f(x)/g(x)? In general, not for this specific function.

mmajames

(g(x)*f'(x)-f(x)*g'(x))/g(x)^2

80past2

okay, so if f(x) = x^2 and g(x) = x^-2, what is the derivative of f(x)/g(x)?

mmajames

f(x) = x^2 and g(x) = x^-2
x^2/x^-2
((x^-2)(2x)-(-2x)(x^2))/ (x^2)^2

80past2

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

I'm mostly having trouble with the second derivatives.

80past2

What do you have so far? Where are you getting stuck?

mmajames

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

Am I doing these correctly?