Hello Math StackExchange,

I am struggling to understand the multivariate chain rule. I have been working on a problem in which I have been given $f=F(x^2 + y^2)$ and I need to find $f_x, f_y, f_{xx}, f_{yy}$.

I have been able to find the first derivatives only when I think about $f$ as the composition of $F(s)$ and $s(x,y)$. Then, it is apparent that $$f_x = \frac{dF}{ds} \frac{\partial s}{\partial x} = 2x\frac{dF}{ds}$$ and $$f_y = \frac{dF}{ds} \frac{\partial s}{\partial y}=2y\frac{dF}{ds}$$.

However, when I go to take the second partial derivatives, I get confused as to whether or not a function depends on a variable or not. I can write the operator: $$f_{xx} = \frac{\partial}{\partial x} \left({2x\frac{dF}{ds}} \right) $$ But, I am confused as to why I must use the product rule in the next step. I think that the notation is making me confused. Ultimately $F'(s)$ is a function of x, so changing $x$ will change the function, correct? As such, the product rule is needed. And $$f_{xx} = 2x\frac{dF_s}{ds} \frac{\partial s}{x} + 2\frac{dF}{ds}$$.

Solving problems of this form has been a struggle in the past. Earlier today, I spent entirely too long trying to find $z_x, z_t, z_{xx}, z_{tt}$ when $z = f(x + t) + g(x-t) $ I had to first realize that the coordinate" z was being used as a function $z(x,y)$ here. Then, taking the first partial derivatives are straight forward for me using the chain rule. However, the second derivatives make much, much less sense. I can see that $$z_{xx} = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial f} \frac{\partial f}{\partial x}+ \frac{\partial z}{\partial g} \frac{\partial g}{\partial x} \right) = \frac{\partial z}{\partial x}\left(z_f f_x + z_g g_x\right).$$ But, like the above problem, isn't $z_f$ ultimately a function of $x$ because $f$ is a function of $x$ as well? Would this require the use of the product rule too?

Is there something that I'm missing in my understanding of multivariable functions, the chain rule, and the product rule? This is all fairly muddy for me, concept-wise. Could I have some help understanding?

Thank you!

up vote 0 down vote accepted

$$f=F(x^2+y^2)$$ $$f_x=\frac{\partial F(x^2+y^2)}{\partial x}=\frac{d F(s)}{d s}\frac{\partial s}{\partial x}$$ $$f_{xx}=\frac{\partial}{\partial x}\bigg(\frac{\partial F(x^2+y^2)}{\partial x}\bigg)=\frac{\partial}{\partial x}\bigg(\frac{d F(s)}{d s}\frac{\partial s}{\partial x}\bigg)=\frac{\partial}{\partial x}\bigg(\frac{d F(s)}{d s}\bigg)\frac{\partial s}{\partial x}+\frac{d F(s)}{d s}\frac{\partial}{\partial x}\bigg(\frac{\partial s}{\partial x}\bigg)$$ $$\Rightarrow f_{xx}=\frac{d^2 F(s)}{d s^2}\frac{\partial s}{\partial x}\frac{\partial s}{\partial x}+\frac{d F(s)}{d s}\frac{\partial^2 s}{\partial x^2}$$ $$\Rightarrow f_{xx}=\frac{d^2 F(s)}{d s^2}4x^2+\frac{d F(s)}{d s}2$$

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