One of the best ways of testing a solution to a difficult problem consists in implementing a different solution, and then comparing the two.

To help test your implementation of derivative, we can use the following property of derivative:

$$

$\backslash$frac$\{\backslash$partial f$\}\{\backslash$partial x$\}\backslash$approx $\backslash$frac$\{$f$($x $+\backslash$Delta$)-$ f$($x$)\}\{\backslash$Delta$\}$

$$

def derivate$\_$approx$($f$,$ x$,$ varval, delta$=0\mathrm{.}0001)$:

$"""$Computes the derivative of f with respect to x$,$ for a given delta,

using the $($f$($x $+$ delta$)-$ f$($x$))/$ delta method. $"""$

# This is f$($x$)$

f$\_$x $=$ compute$($f$,$ varval$=$varval$)$

varval$\_$delta $=$ dict$($varval$)$

varval$\_$delta$[$x$]+=$ delta

# This is f$($x $+$ delta$)$

f$\_$x$\_$plus$\_$delta $=$ compute$($f$,$ varval$=$varval$\_$delta$)$

return $($f$\_$x$\_$plus$\_$delta $-$ f$\_$x$)/$ delta

def similar$($x$,$ y$,$ epsilon$)$:

if x $<0$ and y $<0$:

# If they are negative, max and min play opposite roles.

return similar$(-$x$,-$y$,$ epsilon$)$

if abs$($x $-$ y$)<$ epsilon:

return True

else:

return max$($x$,$ y$)/($min$($x$,$ y$)+$ epsilon$)<1+$ epsilon

#@title Implementation of $`$test$\_$derivative$`$

def test$\_$derivative$($f$,$ df$,$ x$,$ delta$=0\mathrm{.}0001,$ tolerance$=0\mathrm{.}01,$ num$\_$tests$=1000)$:

$"""$Tests if the derivative of f with respect to x is approximately equal to df$.$

Returns True if the test passes for all randomly generated inputs, False otherwise."""

## YOUR SOLUTION HERE

Needs to pass following test:

# Tests $10$ points: $`$test$\_$derivative$`$

f $=("+",("*",$ "cat", "cat"$),("*",$ "dog", "cat"$))$

df$1=("+",("*",2,$ "cat"$),$ "dog"$)$

df$2=("+",("*",2,$ "cat"$),("*",$ "dog", "cat"$))$

assert test$\_$derivative$($f$,$ df$1,$ "cat"$)==$ True

assert test$\_$derivative$($f$,$ df$2,$ "cat"$)==$ False

assert test$\_$derivative$($f$,$ df$1,$ "dog"$)==$ False

assert test$\_$derivative$($f$,$ df$1,$ "donkey"$)==$ False

assert test$\_$derivative$($f$,0,$ "donkey"$)==$ True