I have been trying to get a deeper understanding of calculus and started reading the book "Calculus for the Practical Man". I enjoy the book a lot, but there is a proof about the differential of the square of a variable that I just can't understand. The proof goes like this:
Let $z = mx$ where $m$ is a constant and $x$ is a variable, then $z$ is a dependent variable. Then
$$
z^2 = m^2x^2
$$
We differentiate both sides for both of the equations:
$$
dz = mdx, d(z^2) = m^2d(x^2)
$$
We divide the second results by the first results:
$$
\frac{d(z^2)}{dz} = m * \frac{d(x^2)}{dx}
$$
And then we divide this result by the original equation $z = mx$ to eliminate the constant $m$:
$$
\frac{d(z^2)}{dz} * \frac{1}{z} = \frac{d(x^2)}{dx} * \frac{1}{x}
$$
The conclusion that the author draws from this last part, is that the derivative of the square of a variable divided by the variable itself is the same for any two variables x and z. I understand the rest of the proof, but I just can't understand why the proof until here proves that the derivative of the square of a variable divided by the variable is the same for any two variables. In this proof we used a variable $z$ which was equal to $mx$. As an example I tried to apply the same proof for a variable $z = x + 1$ but could not reduce to the last equality, so I am confused. How about a $z$ which does not depend on $x$ at all, how would that look like? I am a little lost here, so I would appreciate some help. Thanks in advance!