Trouble computing ∂^2 f/∂x^2 (1,1)

[s3a]

Homework Statement

Question:
"Given that the surface x^4 * y^7 + y^6 * z^8 + z^7 * x^9 + 4xyz = 7
has the equation z = f(x,y) in a neighborhood of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

Find:
a) ∂f/∂x(1,1)
b) ∂f/∂y(1,1)
c) ∂^2 f/∂x^2"

Answers:
∂f/∂x(1,1) = -17/19 = -0.894736842105263
∂f/∂y(1,1) = -17/19 = -0.894736842105263
∂^2 f/∂x^2 (1,1) = -2.2399766729844

Homework Equations

Just taking the derivative of
4x^3 * y^7 + 9x^8 * z^7 + 4yz + ∂f/∂x(8z^7 * y^6 + 7z^6 + 4yz).
I also know that ∂f/∂x(1,1) = -17/19.

The Attempt at a Solution

I successfully get every single part of this question except the ∂^2 f/∂x^2 part.

Instead of isolating for ∂f/∂x and trying to differentiate that again with respect to x (which seems very difficult, if not impossible, to do by hand), I just implicitly differentiate for the second time treating ∂f/∂x as a function. I then just plug in -17/19 for ∂f/∂x(1,1) and plug in the point (1,1,1) and get the wrong answer. I tried computing ∂^2 f/∂x^2 several times and keep getting it wrong so any help would be greatly appreciated.

Thanks in advance!

 

[SammyS]

Homework Statement

Question:
"Given that the surface x^4 * y^7 + y^6 * z^8 + z^7 * x^9 + 4xyz = 7
has the equation z = f(x,y) in a neighborhood of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

Find:
a) ∂f/∂x(1,1)
b) ∂f/∂y(1,1)
c) ∂^2 f/∂x^2"

Answers:
∂f/∂x(1,1) = -17/19 = -0.894736842105263
∂f/∂y(1,1) = -17/19 = -0.894736842105263
∂^2 f/∂x^2 (1,1) = -2.2399766729844

Homework Equations

Just taking the derivative of
4x^3 * y^7 + 9x^8 * z^7 + 4yz + ∂f/∂x(8z^7 * y^6 + 7z^6 + 4yz).
I also know that ∂f/∂x(1,1) = -17/19.

The Attempt at a Solution

I successfully get every single part of this question except the ∂^2 f/∂x^2 part.

Instead of isolating for ∂f/∂x and trying to differentiate that again with respect to x (which seems very difficult, if not impossible, to do by hand), I just implicitly differentiate for the second time treating ∂f/∂x as a function. I then just plug in -17/19 for ∂f/∂x(1,1) and plug in the point (1,1,1) and get the wrong answer. I tried computing ∂^2 f/∂x^2 several times and keep getting it wrong so any help would be greatly appreciated.

Thanks in advance!

Here's the Latex of the equation of the surface: $x^4 \, y^7 + y^6 \, z^8 + z^7 \, x^9 + 4xyz = 7$

Here's Latex for your result for the first partial derivative:

$\displaystyle 4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6 + 4yz)=0$​

I got something different:

$\displaystyle 4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6\,x^9 + 4yz)=0$
​

Here's the image of your work, so it's easier to check out.

That's a correct method.

 

[s3a]

I kept getting it wrong so many times but I finally got it right. What I had shown here is wrong for the first differentiation with respect to x and so was yours.

Here is my work if you can read the handwriting and care.

Thank you!

 

[SammyS]

I kept getting it wrong so many times but I finally got it right. What I had shown here is wrong for the first differentiation with respect to x and so was yours.

Here is my work if you can read the handwriting and care.

Thank you!

It took me a long time to see what I had wrong.

Should be 4xy, not 4yz in the parentheses.

Here's the corrected first derivative.

$\displaystyle 4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6\,x^9 + 4xy)=0$

 

[s3a]

Yeah, I saw it. Also, sorry, I should have told you and spared your wasteful search.