Fraction Cancellation Question

Question

I'm trying to learn some maths on my own and have come across this problem.

Find $dy/dx$:

$$ \sqrt{x + y} = \cos(y^2)$$

My working has led me to the below:

$$ -\frac{2\sqrt{x + y}}{2\sqrt{x + y}(1 + 4\sin(y^2)\sqrt{x +y})}$$

Is it legal to cancel out... $$ 2\sqrt{x + y}$$ ...from the top and the bottom?

Answer 1

It is legal to cancel $2\sqrt{x + y}$ because it is being multiplied by all other terms in the denominator. If it was being added or subtracted to something, then you couldn't have canceled it out.

Answer 2

The cancellation is legal because $$ \lim_{y\to\pm x} \frac{\sqrt{x + y}}{\sqrt{x + y}} = 1 $$

Did you end up with $$ \frac{dy}{dx} =-1 -4 \color{blue}{y} \sqrt{x+y} \sin \left(y^2\right)? $$