Partial differentiation of $-(z-x)^4$

Question

Take the derivative of $$-(z-x)^4$$ with respect to $z$

This is my method using the chain rule

$$-4(z-x)^3$$

But the right answer is

$$-4(x-z)^3$$ where $x$ and $z$ have switched signs.

Why so? What happened to make the variables switch sign?

Thanks

Answer 1

Nothing happened, you are right when you say the answer is

$$-4(z-x)^3.$$

The chain rule works perfectly here.

If a book gives a different answer, it is probably a typo.

Edit

Notice that

$$-4(z-x)^3=4(-1)^3(z-x)^3=4(-(z-x))^2=4(x-z)^3.$$

Answer 2

the derivative is given by $$-4(z-x)^3$$

Answer 3

Hint:

The key fact here is that $3$ is an odd exponent, so $-(z-x)^3=(x-z)^3$.