$$f(x)=\frac{3x^2+2\sqrt{x^3+\cfrac{4}{x^4}}}{(x^3-4)\sqrt{x^2+4}}$$
I've thought about this question for a long time but failed to get the answer. How to figure out the substitution $u$ of in order to use the chain rule? Thank you very much.
2026-04-02 11:54:46.1775130886
How to use the chain rule to find the derivative of the function?
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we set $$f(x)=\frac{u}{v}$$ then we have $$f'(x)=\frac{u'v-uv'}{v^2}$$ where $$u=3x^2+2\sqrt{x^3+\frac{4}{x^4}}$$ then we have $$u'=6x+2\frac{1}{2}\left(x^3+\frac{4}{x^4}\right)^{-1/2}\left(3x^2-\frac{16}{x^5}\right)$$ and $$v'=3x^2\sqrt{x^2+4}+(x^3-4)\frac{1}{2}\left(x^2+4\right)^{-1/2}\cdot 2x$$ Can you finish this?