<s>[INST] Chain rule for $ y = (x^2 + x^3)^4$ I am trying to find the derivative of $ y = (x^2 + x^3)^4$
and it seems pretty simple I get
$4(x^2+x^3)^3 (2x+3x^2)$ This seems to be the proper answer to me but the book gets 
$4x^7 (x+1)^3 (3x+2)$ and I have no idea how that happened, what process the author went through or why. My answer seems to be a better and more accurate answer since that is what the chain rule will give you. 
Why is my answer wrong?
 [/INST] You did the differentiation correctly. The book's solution is the same as your answer, but in simplified form. 
To obtain the book's answer from yours: factor $x^2$ from the $(x^2+x^3)$ term in your expression and apply the rule $(ab)^n=a^nb^n$, factor $x$ from the $(2x+3x^2)$ term, and finally combine the $x^6$ and $x$ terms.
$$\eqalign{
4(\color{maroon}{x^2+x^3})^3 (\color{darkgreen}{2x+3x^2})
&= 4  \bigl(\color{maroon}{ x^2(1+x)}\bigr)^3\cdot\color{darkgreen}{ x(2+3x)}\cr 
&=4\cdot (x^2)^3(1+x)^3\cdot x(2+3x)\cr
&=4x^6(1+x)^3\cdot x(2+3x)\cr
&=4x^7(x+1)^3(3x+2).
}
$$
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