<s>[INST] Expressing as a composite function to begin differentiating with the chain rule I'm trying to differentiate $y=(x+1)(x+2)^2(x+3)^3$ using the chain rule, but I am having trouble writing it as a composite function. Any help would be great as I can finish the problem on my own once I just get started in the right direction. Thanks.
 [/INST] When you have products such as $$y=(x+1)(x+2)^2(x+3)^3$$ logarithmic differentiation makes life simpler $$\log(y)=\log(x+1)+2\log(x+2)+3\log(x+3)$$ Differentiating $$\frac{y'}y=\frac 1{x+1}+\frac 2{x+2}+\frac 3{x+1}=\frac {6 x^2+22 x+18 }{(x+1) (x+2) (x+3)}$$ Multiply both sides by $y$ to get $$y'=\frac {6 x^2+22 x+18 }{(x+1) (x+2) (x+3)}(x+1)(x+2)^2(x+3)^3=(6 x^2+22 x+18)(x+2)(x+3)^2$$
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