# Math Help - Implicit differentiation with second derivative

1. ## Implicit differentiation with second derivative

Hi everyone i can't solve this problem:

Find if

(The second derivative can't be expressed solely using .)

I used implicit differentiation an found that DY/DX = -5x/3y
So for i know i have to use the quotient rule but the answer will have y.

Thank you

2. ## Re: Implicit differentiation with second derivative

I think the point of the exercise is that it's OK to have the second derivative in terms of $x$ and $y$. Note. When you have y' in your answer use the first part to eliminate it.

3. ## Re: Implicit differentiation with second derivative

Thank you... yes i subsituted the first derivative but this is what i got:

(-15y-25/y)/(3y)**2

i don't know how to continue. Did i do something wrong??

4. ## Re: Implicit differentiation with second derivative

This is what I got

$y' = -\frac{5x}{3y}$
so

$y'' = -\frac{5}{3} \frac{y - x y'}{y^2}$
Now bring in your first derivative

$y '' = - \frac{5}{3} \dfrac{ y - x \left(- \frac{5x}{3y}\right)}{y^2}$

Now simplify

$y'' = \frac{5}{9} \frac{3y^2+5x^2}{y^3}$

(Notice that you can bring in the original ellipse to simplify a little more!)

5. ## Re: Implicit differentiation with second derivative

thank you but i still don't understand.

I brought down the first derivative and i got:

(-5/3)*[y-x*(-5/3)(x/y)]/(y^2)

But now i don't know how to continue and simplify this. If you could explain it to me

Thanks

6. ## Re: Implicit differentiation with second derivative

clear the complex fraction by multipying by $\frac{3y}{3y}$ ...

$-\frac{5}{3} \cdot \frac{y + \frac{5x^2}{3y}}{y^2} \cdot \frac{3y}{3y}$