Suppose $\phi(x)$ is defined by a formula in which x occurs in several places (for example, there are three $x$’s in $\phi(x)= > \frac{x^2e^x}{x+3}$).
Show that the derivative $\phi'(x)$ is obtained by differentiating with respect to each of the $x$’s in turn, treating the others as constants, and adding the results.
Hint: Let $F(x_1, . . . , x_n)$ be the function of several variables obtained by replacing each of the $x$’s in the formula for $\phi(x)$ by a different variable.
I need to Express $\phi$ in terms of F and use the chain rule. Now i am having trouble in proving that $\phi'(x)$ is actually the sum of partial derivatives of the function. And therefore cant really reach to a conclusion for this question. The answer in the text book was given the following
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The multivariable chain rule says that if you assume each variable $x_i = x_i(t)$ is a function of another variable $t$, then the total derivative is
$$ \frac{dF}{dt} = \sum_i \frac{\partial F}{\partial x_i} \frac{dx_i}{dt} $$
If you take $x_i(t) = t$ for all $i$, then $\frac{dx_i}{dt} = 1$, and so you get that $\frac{dF}{dt}$ is the sum of the partial derivatives (with $x_i = t$ substituted).