I am learning chain rule for two variable functions and in Stewart's book i have the exercise:

If $z=f(x, y)$ and $x=r^2 + s^2, y = 2rs$ then get $\frac{\partial ^2z}{\partial r^2}$

And i can get $\frac{\partial z}{\partial r}$, but in development of the exercise in Stewart's book a term appears that I do not know where it comes from.

Where does the term $2\frac{\partial z}{\partial x}$ come from?

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Answering question directly: it is first member from formula $(f\cdot g)'=f'g+g'f$. In your case $$\frac{\partial}{\partial r}\left(2r \cdot\frac{\partial z}{\partial x}\right)=\frac{\partial}{\partial r}2r \cdot \frac{\partial z}{\partial x}+ 2r \cdot\frac{\partial}{\partial r}\frac{\partial z}{\partial x} $$ and $\frac{\partial}{\partial r}2r = \frac{d}{d r}2r=2$

  • $\begingroup$ ohh, my bad. Thanks =) $\endgroup$ May 26 at 0:34
  • $\begingroup$ Glad to be helpful . $\endgroup$
    – zkutch
    May 26 at 0:36

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