finding derivative

**droidus** · June 13th 2012, 11:24 AM

I need help finding the derivative of this. I am a bit confused. I have y = (x^2)/(x^2+3). the derivative is 6x/(x^2+3)^2. I understand the quotient rule, and how they got the squared on the bottom. but how did they get the 6x on the top?

**Reckoner** · June 13th 2012, 11:51 AM

Originally Posted by droidus

I need help finding the derivative of this. I am a bit confused. I have y = (x^2)/(x^2+3). the derivative is 6x/(x^2+3)^2. I understand the quotient rule, and how they got the squared on the bottom. but how did they get the 6x on the top?

Next time show your work. You probably made a mistake somewhere...

$y = \frac{x^2}{x^2+3}$

$\frac{dy}{dx} = \frac{\left(x^2+3\right)(2x)-x^2(2x)}{\left(x^2+3\right)^2}$

$= \frac{2x^3+6x-2x^3}{\left(x^2+3\right)^2}$

$= \frac{6x}{\left(x^2+3\right)^2}$

**droidus** · June 13th 2012, 01:14 PM

ok - here is what i get for the second derivative then...

[(x^2 + 3)^2 + 6 - 6x(4x)] / (x^2+3)^4

**Reckoner** · June 13th 2012, 01:29 PM

Originally Posted by droidus

ok - here is what i get for the second derivative then...

[(x^2 + 3)^2 + 6 - 6x(4x)] / (x^2+3)^4

The derivative of $\left(x^2+3\right)^2$ is not $4x.$ And shouldn't you be multiplying that 6 instead of adding it? Let's be more careful:

$\frac{dy}{dx} = \frac{6x}{\left(x^2+3\right)^2}$

$\frac{d^2y}{dx^2} = \frac{\left(x^2+3\right)^2\cdot\frac d{dx}[6x] - 6x\cdot\frac d{dx}\left[\left(x^2+3\right)^2\right]}{\left(x^2+3\right)^4}$

$= \frac{6\left(x^2+3\right)^2 - 6x\cdot2\left(x^2+3\right)\cdot\frac d{dx}\left[x^2+3\right]}{\left(x^2+3\right)^4}$

$= \frac{6\left(x^2+3\right)^2 - 24x^2\left(x^2+3\right)}{\left(x^2+3\right)^4}$

$= \frac{6\left(x^2+3\right) - 24x^2}{\left(x^2+3\right)^3}$

$= \frac{-18\left(x^2-1\right)}{\left(x^2+3\right)^3}$

$= \frac{-18(x-1)(x+1)}{\left(x^2+3\right)^3}$

Make sure that you're using the chain rule properly. We must remember to differentiate the "inner" function after we differentiate the "outer" function of a composite function.

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