Second Derivative using Implicit Differentiation

[Bear_B]

Homework Statement

Find y''(x) of the parametric equation 9x^2+y^2=9 using implicit differentiation.

Homework Equations

I already came up with y'(x) = -9x/y

The Attempt at a Solution

Here is what I have for y''(x) so far

y''(x) = d/dx (-9xy^-1)
=-9(d/dx)(xy^-1)
=-9(x(d/dx)(y^-1)+(y^-1)(dx/dx))
=-9(-x(y^-2)y'(x)+y^-1)

I substituted the value of y'(x) = -9x/y here

=-9((-x/(y^2))(-9x/y)+(1/y))
=-9((9x^2)/(y^3)+(1/y))
=(-81x^2)/(y^3)-(9/y)

I know this is incorrect, I originally tried this using the quotient rule but was getting the same answer and the work was much more jumbled, so I opted for the product rule.

The book states the answer is -81/(y^3). I am stuck and haven't been able to work towards the right answer.

 

[sponsoredwalk]

Use the quotient rule after you take the first derivative, then look for any possible substitutions that you think might work (& don't be afraid of fractions on fractions on fractions lol).

When you've got it as clean as you can get it (it isn't a messy equation but be careful w/ minus signs) look to see if multiplying the equation (i.e. top & bottom) by a clever choice of 1 will help.

let us know how it goes.

 

[vela]

You're good so far. Put your result over a common denominator and simplify the top.

 

[Bear_B]

Ok, I got the answer. The tip on substitution was what really allowed my to break this one open. Since I think this is a great problem that requires a creative strategy (and possibly multiple attempts) to get the simplest answer, I am going to post the rest of my work here.

The result of the Product or Quotient Rule and the substitution for y'(x) yields:

y''(x)=(-9y-((81x^2)/y))/(y^2)

From the original equation, x^2=1-(y^2)/9
I used this value to substitute for x^2.

=(-9y-81((1-(y^2)/9)/(y^2)))/(y^2)

There are a lot of fractions going on, but from here it is just a matter of reducing and simplification.

=(-9y-81(1/y-(y^2)/9y))/(y^2)
=(-9y-81/y+81y/9)/(y^2)
=(-9y-81/y+9y)/(y^2)
=(-81/y)/(y^2)
=-81/y*1/(y^2)
y''(x)=-81/(y^3)

Thanks for the prompt tips. Your feedback helped me solve this problem and find the simplest way to express y''(x).
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