Problem Suppose $\varphi(x)$ is defined by a formula in which $x$ occurs in several places. (e.g. $\varphi(x) = \dfrac{x^2e^x}{x+3}$.)
1) Show that the function's derivative is obtained by differentiating with respect to each of the x's in turn, treating the others as constants, and adding the results.
2) Notice the rules for differentiating sums and products are special cases of this result, obtained by taking $\varphi(x) = f(x) + g(x)$ or $\varphi(x) = f(x)g(x)$. What is the derivative of $\varphi(x) = f(x)^{g(x)}$?
The hint says
If the variable occurs in $n$ places in the formula for $\varphi$, let $F(x_1,\dots,x_n)$ be the function of n variables obtained by replacing each of the $x$'s in the formula by a different variable.
My thought for (1) is to take the partial derivative of $\varphi(x)$ with respect to each $x_j$ using chain rule, but I am not sure how to write down the expression. I have no clue how to approach (2). Any help is appreciated!
You need a second function $\psi(x)=(x, ..., x)$ with $n$ copies of $x$. Now for example if $\varphi(x)=f(x)^{g(x)}$, we have $F(x_1, x_2)=f(x_1)^{g(x_2)}$ and $\varphi(x)=F(\psi(x))$. Now you can apply the chain rule and compute the derivative of $\varphi$.