<s>[INST] Derivative of $f(x) = (x^2 +1)^3 (2x+5)^2$ I have function
$$f(x) = (x^2 +1)^3 (2x+5)^2$$
I need to find the derivative.
I believe that I need to use the product rule and chain rule.
Here's what I did.
$$f'(x) = (2x+5)^2[3(x^2+1)^2(2x)] + (x^2+1)^3[2(2x+5)2]
\\ = (2x+5)^2(6x(x^2+1)^2)+4(2x+5)(x^2+1)^3
\\ = 6x(2x+5)^2(x^2+1)^2+4(2x+5)(x^2+1)^3$$ 
That is what I have.
But the problem is, the answer book says that the answer is,
$$2(x^2+1)^2(2x+5)(8x^2+15x+2)$$
Since the answer book has been wrong few times, so I checked the answer with a calculator.
The calculator showed the same answer.
Then I used another calculator (which showed some steps), showed my answer.
The calculator showed, 
Which answer is correct?? Or are they just equivalent answers??
If I did wrong, what was the problem??
Can it be factorized? How?
Thank you
 [/INST] Factor out common terms:
$$ 6x(2x+5)^2(x^2+1)^2+4(2x+5)(x^2+1)^3 = 2(2x+5)(x^2+1)^2(3x(2x+5) + 2(x^2 + 1))$$
Simplify to get $$2(2x+5)(x^2+1)^2(6x^2+ 15x + 2x^2 + 2)= 2(2x+5)(x^2+1)^2(8x^2+ 15x + 2)$$
The text book is correct, AND your work was fine (take comfort in that). It was just a matter of simplifying your work through factorization. 
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