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Say I was given an implicit function $$x^4+ y^3+2xy=6$$ My textbook says I should treat $y^3$ as a composite function and $2xy$ as a product. My question is why can't $x^4$ be treated as a composite function expressed in terms of $x$? I mean: $$f(x)=x^2$$ $$g(u)=u^2$$ Therefore, $$(g(f(x))=x^4 $$ So why should $y^3$ only be treated as a composite function when $x^4$ is also a composite function just expressed in terms of $x$?

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    $\begingroup$ This is where algebra departs from science and becomes an art. You are free to be creative and maybe it will lead you to new discoveries. $\endgroup$ Feb 21, 2019 at 16:53
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Anything can be viewed as a composition of any number of functions, but it's easier if you use the smallest number of functions possible. So while you can compute

$$\frac{d}{dx} x^4 = \frac{d}{dx} (x^2)^2 = 2(x^2) \frac{d}{dx} x^2 = 2(x^2)(2x) = 4 x^3$$

it's easier to just use the power rule and get this answer directly.

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Actually, every function can be viewed as a composite function.

As an example, consider this trivial case. Let $f(x)$ be any function, and let $g(x)=x$. Then we can say that $$f(x)=g(f(x))=f(g(x))$$

I don't know why your textbook did not mention that $x^4$ is a composite function. My best guess would be that their focus was on $y$ and that they did not define any other functions like the function $f(x)=x^2$ you mentioned.

So, your reasoning is correct.

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