What is dy/dx if y = (x^2 + 2)^3 (x^3 + 3)^2and explain how you got your answer. i have no knowledge on how to get the answer at all.

We're given the following equation:


In order to find  we must differentiate both sides of the equation.

Lets start differentiating the left side ():


The  simply servers to let us know we're differentiating whatever follows it (in this case ) with respect to .

What we used to get the result is called the "power rule for differentiation" it states the following:


In which is any variable (in the previous case ) and is any constant (in the previous case this ).

Now we'll differentiate the right side of the equation ():


What we did to differentiate the right side was, first, apply something called "product rule" for differentiation, it states the following:
 

In which  and are arbitrary functions of an arbitrary variable () (in this case and ).

After that we applied something called "chain rule" for differentiation, which states the following:
if , then


Finally, the  we introduced as a factor after differentiating the right side (we also did it with the left side but with a ) is a consequence of the chain rule, it is always done.

Finally, equaling both differentiated sides of the equation we have:


We solve for , and the answer is:

What’s the improper fraction?