Derive this: (x^2 + 2ax - a) / ((x + a)^2)

[spacejunkie]

I just started CALC II and I'm woefully behind already. Hopefully I'll be able to catch up this weekend but I have a mental block with a particular derivative. Will someone please help?

(x^2 + 2ax - a) / ((x + a)^2)

The answer is:

(2a(a + 1)) / ((x + a)^3)

But how? :?

I've tried to solve this several times and just can't get it what am I missing?

 

[royhaas]

The numerator can be rewritten as \(\displaystyle (x+a)^2-(a+a^2)\).

 

[stapel]

By "derive", do you mean "differentiate"? So you're supposed to differentiate the first expression, and you've been given that the second expression is the simplified derivative?

Thank you.

Eliz.

 

[spacejunkie]

By "derive", do you mean "differentiate"? So you're supposed to differentiate the first expression, and you've been given that the second expression is the simplified derivative?

Thank you.

Eliz.

Yes you've got it right.

 

[stapel]

Yes you've got it right.

Great! So plug this into the Quotient Rule, and simplify the result.

. . . . .Quotient Rule: Given that f(x) = g(x)/h(x), the derivative is:

. . . . . . . .f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]²

. . . . .Let g(x) = x² + 2ax - a and h(x) = (x + a)²....

If you get stuck, please reply showing all of your steps. Thank you.

Eliz.

 

[spacejunkie]

((x+a)^2 dy/dx (x^2+2ax-a) - (x^2+2ax-a) dy/dx (x+a)^2) / ((x+a)^4) =

((x+a)^2 (2x+2a) - (x^2+2ax-a) (2x+2a)) / ((x+a)^4) =

((x+a)^2 2(x+a)) / ((x+a)^4) - ((x^2+2ax-a) 2(x+a)) / ((x+a)^4) =

(2(x+a)^2) / ((x+a)^3) - (2(x^2+2ax-a)) / ((x+a)^3) =

(2(x+a)^2 - 2(x^2+2ax-a)) / ((x+a)^3) =

That's as far as I was getting.

Now, using royhaas' hint, I'm getting:

(2(x+a)^2 - 2(x+a)^2 + (a+a^2)) / ((x+a)^3) =

(a+a^2) / (x+a)^3) = Not the answer. Still off by a factor of 2. Where is that damn 2? :?

Sorry for the text. I can't download the math text editor on this computer.

 

[stapel]

(2(x+a)^2 - 2(x^2+2ax-a)) / ((x+a)^3) =

That's as far as I was getting.

It looks good so far. (Thank you for typing that all out!) Now let's see about that numerator:

. . . . .2(x + a)² - 2(x² + 2ax - a)

. . . . .2(x² + 2ax + a²) - 2(x² + 2ax - a)

. . . . .2x² + 4ax + 2a² - 2x² - 4ax + 2a

. . . . .2x² - 2x² + 4ax - 4ax + 2a² + 2a

. . . . .2a² + 2a

Factor.

Eliz.

 

[spacejunkie]

Doe!

Thank you so much!

Cheers,
The Spacejunkie

 

[stapel]

Glad I could help!

Eliz.