Question: I have been doing differential calculus on Khan Academy for some time. Sometimes when Mr Khan decides to avoid the chain rule (because there is a sqaure root in the function), instead of working with square roots, he decides that S(x) (which is our function for this question) is equal to (F(x))^2 which effectively gets rid of the square root so we can avoid doing the chain rule in order to find our derivatives. (See the link below)

![Picture of Equation](https://i.stack.imgur.com/o08Fu.jpg)

What I dont understand is how this is not changing the original function into a completely different function with different answers.

Thank you so much for your help.

Regards,

Joshua

Answer: 
This is in the context of max/min problems. What is being used here is that if $f$ is a *nonnegative* function, then maximizing its square is the same as maximizing/minimizing the function. The square root is the principal square root, i.e. the positive one, so here this applies.

The reason for this is that the function 
$$
x\mapsto x^2
$$
is increasing for nonnegative $x$, so it maintains inequalities, namely 
$$
x\geq y\iff x^2\geq y^2
$$
And of course the same holds for the opposite inequality sign. 

Since Sal is only human, he would prefer to take the derivative of something without a radical than with.
When you square a function, all positive maxima and negative minima become maxima, while all negative maxima, positive minima and zeroes become minima.

[![enter image description here](https://i.stack.imgur.com/72G2b.png)](https://i.stack.imgur.com/72G2b.png)