How Do You Find the First and Second Derivatives of These Functions?

[mmajames]

Homework Statement

Just trying to find the first and second derivatives.

X^2/(X^2-16)

1+X/1-X

X^3(X-2)^2

Homework Equations

Quotient Rule/Power Rule/Chain Rule

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?