# Thread: Derivative of (x*y) / (2+x)(4+y)

1. ## Derivative of (x*y) / (2+x)(4+y)

Find the partial derivative of (x*y) / (2+x)(4+y) with respect to x...

I know the answer is supposed to be

2y / (4+y)(2+x)^2

I have used Maple's tutor function, but still don't get it... I think I need to know what rules to apply and do I do anything to the equation before I start to differentiate?

Simon DK, thanks.

2. Originally Posted by sh01by
Find the partial derivative of (x*y) / (2+x)(4+y) with respect to x...

I know the answer is supposed to be

2y / (4+y)(2+x)^2
WRT $\displaystyle x.$ So $\displaystyle y$ behaves like a constant. Then we're just interested on compute the derivative of

$\displaystyle \frac{x} {{x + 2}} = \frac{{x + 2 - 2}} {{x + 2}} = 1 - \frac{2} {{x + 2}}.$

The derivative of this function is $\displaystyle \frac2{(x+2)^2}.$ The conclusion follows.

3. Hello, sh01by!

Find the partial derivative with respect to x: .$\displaystyle f(x,y) \;=\;\frac{xy}{(x+2)(y+4)}$
Answer: .$\displaystyle \frac{2y}{(x+2)^2(y+4)}$
Quotient Rule . . .

. . $\displaystyle \frac{\partial f}{\partial x} \;=\;\frac{\overbrace{(x+2)(y+4)}^{\text{den}} \cdot \overbrace{y}^{\text{(num)'}} \;- \;\overbrace{(xy)}^{\text{num}} \cdot\overbrace{(y+4)\cdot1}^{\text{(den)'}} } {\underbrace{(x+2)^2(y+4)^2}_{\text{(den)}^2} }$

Factor: . $\displaystyle \frac{(y+4)\cdot[y(x+2) - xy]}{(x+2)^2(y+4)^2} \;=\;\frac{xy + 2y - xy}{(x+2)^2(y+4)} \;=\;\frac{2y}{(x+2)^2(y+4)}$

Wow! . . . I love your method, Krizalid!

4. ## Thanks

Thanks! I tried using the formula and more or less found the same answer, but the trouble was the factorization. It amounts to much trouble, if you don't know the truly important rules of factorization.

Thanks Soroban -- and Krizalid thanks to 'u too for 'ur effort. :o)

5. ## Taking the derivative of the derivative

Okay, so how do I do the factoring, if I wish to take the derivative of the derivative with respect to x?

I know the solution is

-4/((2+x)^3(4+y) or as seen on http://www.econ.ku.dk/polit/studeren..._Mat2004_i.pdf (however, in Danish)... But math is universal, right? :o)

Simon DK.

I ended up with (-4x^2 + 16)/((2+x)^4(4+y)) ?