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 (

):
![\frac{d}{dx}(x^2+2)^3(x^3+3)^2=[6x(x^2+2)^2(x^3+3)^2+6x^2(x^3+3)(x^2+2)^3]dx](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%28x%5E2%2B2%29%5E3%28x%5E3%2B3%29%5E2%3D%5B6x%28x%5E2%2B2%29%5E2%28x%5E3%2B3%29%5E2%2B6x%5E2%28x%5E3%2B3%29%28x%5E2%2B2%29%5E3%5Ddx)
What we did to differentiate the right side was, first, apply something called "product rule" for differentiation, it states the following:
![\frac{d}{ds}[f(s)g(s)]= \frac{d}{ds}[f(s)]g(s)+\frac{d}{ds}[g(s)]f(s)](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bds%7D%5Bf%28s%29g%28s%29%5D%3D%20%5Cfrac%7Bd%7D%7Bds%7D%5Bf%28s%29%5Dg%28s%29%2B%5Cfrac%7Bd%7D%7Bds%7D%5Bg%28s%29%5Df%28s%29%20)
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
![\frac{d}{ds}[h(s)]= \frac{d}{ds}[g(f(s))] \frac{d}{ds}[f(s)]](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bds%7D%5Bh%28s%29%5D%3D%20%5Cfrac%7Bd%7D%7Bds%7D%5Bg%28f%28s%29%29%5D%20%5Cfrac%7Bd%7D%7Bds%7D%5Bf%28s%29%5D%20%20%20)
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: