# Thread: ODEs differentiating y with respect to x

1. ## ODEs differentiating y with respect to x

What is the derivative with respect to x of . . . . xy'/y^2
I can get as far as derivative of xy' = y' + xy"
But I dont know what to do about the bit on the bottom . .
I'm guessing i need the quotient rule somehow . .

Cheers

2. Originally Posted by eatmoreburritos
What is the derivative with respect to x of . . . . xy'/y^2
I can get as far as derivative of xy' = y' + xy"
But I dont know what to do about the bit on the bottom . .
I'm guessing i need the quotient rule somehow . .

Cheers
You have to use the quotient rule: The derivative of $\frac{u}{v}$ is equal to $\frac{\frac{du}{dx} \, v - \frac{dv}{dx} \, u}{v^2}$.

$u = xy' \Rightarrow \frac{du}{dx} = y' + x \, y''$, as you correctly found (using the product rule).
$v = y^2$. To get $\frac{dv}{dx}$ you have to use the chain rule: $\frac{dv}{dx} = 2y\, y'$.