autodiff test framework is not testing first gradients

Description

Add this code to stan/math/fwd/fun/inv.hpp:

template <typename T>
inline fvar<fvar<T>> inv(const fvar<fvar<T>>& x) {
  T x_sq = square(x.val_.val_);
  fvar<T> first(1 / x.val_.val_, 100.0); // incorrect, second argument should be -x.val_.d_ / x_sq
  fvar<T> second(-x.d_.val_ / x_sq, -x.d_.d_ / x_sq + 2 * x.val_.d_ * x.d_.val_ / (x.val_.val_ * x_sq));
  return fvar<fvar<T>>(first, second);
}

./runTests.py test/unit/math/mix/fun/inv_test.cpp will pass even though the first gradient is wrong.

I'm attaching a rendered copy of the Nomad manual cause that's where I'm getting the eqs and terminology and such (check table 2):
autodiff.pdf

Expected Output

This should throw an error when there is a bug.

Current Version:

v3.4.0

Start with stan/math/fwd/fun/inv.hpp and test/unit/math/mix/fun/inv_test.cpp. Run ./runTests.py test/unit/math/mix/fun/inv_test.cpp to inspect why the incorrect first gradient passes. Done means the test throws an error when the first gradient is wrong.