# Identifying the composite function(s) in an implicit function

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 $$x$$?

• 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. Feb 21, 2019 at 16:53
• Has your question been answered? If yes, you should accept an answer. Jun 24, 2019 at 15:01

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.

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.