Help with finding the Derivative?

Question

I am trying to use the quotient rule, but I just can't seem to understand it. The derivative I am trying to find is crucial because it is the 2nd derivative in my problem here...

Please help me find the derivative of:

-20x/(x^2-9)^2

Thanks...

So, I need help finding the derivative of:

-20x/(x^2-9)^2

**in simplified form**

Right now, I am getting 80x^2(x^2-9)/(x^2-9)^3

Can I get a quick check from someone?

Answer 1

Ok, so if you have f(x)/g(x), quotient rule is [g(x)f'(x)-g'(x)f(x)]/[g(x)]2

so your f'(x) = -20
and for g'(x) you have to use chain rule, so g'(x) = 2(x^2-9)*2x
or 4x(x^2-9)

so the derivative of the whole thing is:

([(x^2-9)^2 * -20] - [4x(x^2-9)*-20x])/(x^2-9)^4

= [-20(x^2-9)^2 + 80x^2(x^2-9)]/(x^2-9)^4

you can factor out an (x^2-9) from both the numerator and denominator and cancel it
leaving you with

= [-20(x^2-9) + 80x^2]/(x^2-9)^3

if you then multiply out the numerator, you get

= (60x^2 + 180)/(x^2-9)^3

If you want, you can use pascal's triangle to multiply out the deniminator, but I don't really think that's necessary.

Answer 2

The quotient rule is (LO d(HI) - HI d(LO))/(LO^2)

So the derivative here is:

[ (x^2 - 9)^2 * (-20) - (-20x) * 2(x^2 - 9) * (2x) ] / (x^2 - 9)^4

= 20(x^2 - 9)(9 - x^2 + 4x^2) / (x^2 - 9)^4 = 60(x^2 + 3) / (x^2 - 9)^3

If you don't remember the quotient rule, and do remember the chain rule, you can do this by taking logs (to be totally rigorous, you have to split into when the function is positive or negative, but we'll ignore that). If you don't get what's going on here, you might not have learned it yet (so ignore it for now).

Let y = -20x/(x^2 - 9)^2.

Then log (y) = log(-20) + log(x) - 2log(x^2 - 9). So taking the derivative of each side

dy/y = dx/x - 4xdx/(x^2 - 9)

So dy/dx = y * [ 1/x - 4x/(x^2 - 9) ] = y * (x^2 - 9 - 4x^2) / x(x^2 - 9)

So dy/dx = -3y(x^2 + 3) / x(x^2 - 9) so now plugging in y

dy/dx = 60(x^2 + 3) / (x^2 - 9)^3

Answer 3

f(x)=-20x/(x^2-9)^2
f'(x)= [20(x^2-9)^2+80x^2(x^2-9)] / [(x^2-9)^4]