# Following functions: f(x_1, x_2)=[x_1x_2^2+x^3_1x_2 x^2_1x_2+x_1+x^3_2] g(u)=[e^u u^2+u] Is it possible to take a derivative of f(g) or g(f). If not - why?

Following functions:
$f\left({x}_{1},{x}_{2}\right)=\left[\begin{array}{c}{x}_{1}{x}_{2}^{2}+{x}_{1}^{3}{x}_{2}\\ {x}_{1}^{2}{x}_{2}+{x}_{1}+{x}_{2}^{3}\end{array}\right]$
$g\left(u\right)=\left[\begin{array}{c}{e}^{u}\\ {u}^{2}+u\end{array}\right]$
Is it possible to take a derivative of $f\left(g\right)$ or $g\left(f\right)$. If not - why?
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Zackary Galloway
$f:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ and $g:\mathbb{R}\to {\mathbb{R}}^{2}$.
Both functions are of class ${C}^{\mathrm{\infty }}$ on their domains, so we only need to know how composing the two functions makes sense. We can't compose $g$ with $f$, but we can compose $f$ with $g$ and $f\circ g:\mathbb{R}\to {\mathbb{R}}^{2}$: it is a vector function, and $\left(f\circ g{\right)}^{\prime }\left(u\right)={J}_{f}\left(g\left(u\right)\right){g}^{\prime }\left(u\right)$, where ${J}_{f}$ is the Jacobian matrix of $f$.