Why Does an Extra Factor Appear in My Calculus Chain Rule Proof?

[laser1]

Homework Statement

See description

Relevant Equations

Chain Rule

Part (a) is fine. I have attached my work for this here anyway because I use the first two lines of (a) in my proof of (b). In part (b), I get the extra factor of $\frac{1}{r} \frac{\partial z}{\partial r}$ and I don't know why. Can anyone help? I am unable to figure out the flaw in my steps :(

 

[renormalize]

Please learn LaTeX and post equations using that format. Check out the handy guide at the bottom left of this post.

 

[laser1]

Please learn LaTeX and post equations using that format. Check out the handy guide at the bottom left of this post.

ok

 

[renormalize]

Sorry, I don't see any difference between the problem-statement equation (top box) and your derived equation (bottom boxes):

 

[laser1]

Sorry, I don't see any difference between the problem-statement equation (top box) and your derived equation (bottom boxes):
View attachment 354357

my bad I have edited #3

 

[PeroK]

my bad I have edited #3

The partial derivative of $z$ wrt $x$ is a function of both $x$ and $y$. The second order derivative of $z$ wrt $r$ should have terms like $\frac{\partial^2 z}{\partial x \partial y}$.

Try $f(x,y) = xy$ as an example to check your working.

 

[PeroK]

That said, the cross terms cancel. But, you are also missing the terms arising from the product rule when taking the second derivative wrt $\theta$.

 

[laser1]

That said, the cross terms cancel. But, you are also missing the terms arising from the product rule when taking the second derivative wrt $\theta$.

Good catch, it all works out now, thanks!