[Community Question] Calculus: If $g(x)$ be the inverse of $f(x)$ then prove that $2g''=3g^2$

One of our user asked:

Let $$f(x)=\int_0^x\frac{dt}{\sqrt{1+t^3}}$$ Prove that $$2g''=3g^2$$ given $g(x)$ is inverse of $f(x)$.

I tried of applying Newton-Leibnitz both sides but could not succeed as the variable is $x$ on the left and $t$ on the right.

How to do this?

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