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I have this function I wish to use in composing a Jacobian. I've simplified it as follows: $$f(x,y) =-A \sqrt{|x + y|}\ log(B + \frac{C}{\sqrt{|x + y|}})$$ So, I need to calculate the partial derivatives.

After some refresher study it appears I need to use the chain rule and product rule. So, let $H(x,y) = \sqrt{|x + y|}$

Then the derivative is
$$\frac{df}{dx}=-A[\frac{dH}{dx}log(B+\frac{C}{H}) + H\frac{d}{dx}log(B+\frac{C}{H}]$$ $$=-A[\frac{sign(x)sign(x+y)}{2H}log(B+\frac{C}{H})+\frac{H}{(B+\frac{C}{H})ln(10)}\frac{d}{dx}(B+\frac{C}{H})]$$ $$\frac{d}{dx}(B+\frac{C}{H})=\frac{-C}{2}|x+y|^{-3/2}sign(x)sign(x+y)$$ so $$\frac{df}{dx}=\frac{-A}{2H}sign(x)sign(x+y)[log(B+\frac{C}{H})-\frac{C}{(B+\frac{C}{H})ln(10)})]$$ is this correct?
I haven't done this since 1988, so I may have missed something or lack the notation you prefer.
thanks much!

Trenton J
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  • I am struggling with the notation. Can we please use $x$ and $y$ instead of $h_1$ and $h_2$? Then your starting point is a function $f(x,y)$ which would be a straightforward function of two variables except for including absolute value $|x-y|$. Where I am really stuck is that you introduce $h$ without defining it. Is $h=x-y$? Then in your next line you only consider one of the two partial derivatives. – AlgTop1854 Oct 24 '23 at 19:43

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