91Ó°ÊÓ

The objective is to prove that the chain rule in chapter 2 is special case and to explain why

For a given function, $z=f(x_1,x_2....,x_n)$is the complete version of chain rule. $x_i=u_i(t_1,t_2,...t_m)$and $t_1,t_2,...t_n$for all values at which each $u_i$is differentiable and if $f$is differentiable at $1≤I≤n$

$\left(x_1,x_2,…,x_n\right)$then

$\frac{∂z}{∂t_j}=\frac{∂z}{∂x_1}\frac{∂x_1}{∂t_j}+\frac{∂z}{∂x_2}\frac{∂x_2}{∂t_j}+…+\frac{∂z}{∂x_n}\frac{∂x_n}{∂t_j}……$

where $1≤j≤m$.

For a single variable function, the chain rule is

$\frac{dz}{d{t}_1}=\frac{dz}{d{x}_1}\frac{d{x}_1}{d{t}_1}$

The goal is to demonstrate that for $n=m=1$, the complete version of chain rule yields a single-variable chain rule.

The complete version of the chain rule is as follows when $n=m=1$. $z=f\left(x_1\right)$and $x_i=u_i\left(t_1\right)$for a given function $n=m=1$in place of $1≤i$for the values of $t_1$at which $u_1$is differentiable and if $f$is differentiable at $x_1$.

$\frac{dz}{d{t}_1}=\frac{dz}{d{x}_1}\frac{d{x}_1}{d{t}_1}$ [Because the function has only one variable, the partial derivative is identical to the normal derivative.]

Hence proved.