Taking $\frac{\text{d} (xy)}{\text{d} (xy)}$ as a starter example, I'd say with reasonable confidence that this is 1. Turning it up a bit, for example $\frac{\text{d} (xy)^2}{\text{d} (xy)}=2xy$ also makes sense to me. But what if it gets even more complicated? Take $\frac{\text{d} (x^3+xy^2)}{\text{d} (x+xy)}$, I wouldn't have any idea how to tackle this one, but I'm thinking as both values are clearly defined and depend on each other, so in a sense the derivative should exist as well? Do these expression make any mathematical sense, and can they actually be evaluated? Where's the limit?
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In these multivariate examples you may have to use chain rule, but if it is just one variable the best way to do it is either redefine the expression or use the chain rule, e.g. $$\frac{\mathrm{d}(x^2)}{\mathrm{d}(\sqrt{x})}=\frac{\mathrm{d}(\sqrt{x}^4)}{\mathrm{d}(\sqrt{x})}$$ or: $$\frac{\mathrm{d}(x^2)}{\mathrm{d}(\sqrt{x})}=\frac{\mathrm{d}(x^2)}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}(\sqrt{x})}=\frac{\mathrm{d}(x^2)}{\mathrm{d}x}\left(\frac{\mathrm{d}(\sqrt{x})}{\mathrm{d}x}\right)^{-1}$$ as for examples like you stated, use the fact that: $$u=u(x,y),x=x(t),y=y(t)\\ \frac{\mathrm{d} u}{\mathrm{d} t}=\frac{\partial u}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t}+\frac{\partial u}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t}$$
Henry Lee
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