Part $3$: The Chain Rule

$3\mathrm{.}1[2$ pt$]$ Use Sympy to make two functions $y=x^3-x$ and $z=\sqrt[\phantom{2}]{y}+5.$ You will have to use different variables for the output $y($in the first function$)$ and the input $y($in the second function$)$; please use ${?}_y$ and $y,$ respectively. Set the variables $d{y}_{d}x$ and $dz$ with respect to $x$ and the derivative of $z$ with respect to $y,$ respectively.

Next, use those two variables and the chain rule to define the variable $dz\_$ dx$\_$no$\_$sub, which is the derivative of $z$ with respect to $x.$ It should contain that variables $x$ and $y.$ To connect the notation here to the notation in the slides, ${?}_y=g(x)$ and $z=f(y).$ Finally, substitute the variable ${?}_y$ in for the variable $y$ in the function $d{z}_{-}d{x}_{-}$no$\_$sub in order to define $dz\_dx\_$ with$\_$sub. It should only contain the variable $x.$

$[39]$:

import sympy as sp

from sympy import $*$

import math

from math import $*$

$x,y=$ sp$.$symbols $(\text{'}($:$x_y^{\text{'}}\}$

$y=x****3-x$

$z=sp.$ sqrt ${(}_y)+5$

$d{y}_{d}x=sp*diff(y,x)$

$d{z}_{d}y=sp*diff(z,$dotsy$)$

$d{z}_d{x}_n{o}_{s}ub=d{z}_{d}y**d{y}_{d}x$

$dz\_$dx$\_$with$\_$sub $\frac{?}{b}=ar(dz)\_$dx$\_$no$\_$sub.subs ${(}_y,y)$