<s>[INST] How to get first derivative? My question is to find the first derivative using product and/or quotient rule
$$f(x) = {(x^2+1)(x^2-2) \over 3x+2}.$$
The solution to the problem is:
 $$f'(x) = {[(2x)(x^2-2)+(x^2+1)(2x)](3x+2)-(3)[(x^2+1)(x^2-2)] \over (3x+2)^2}.$$
I'm having problems getting the same solution would someone be able to help me by showing me the correct way of getting the first derivative of this question
 [/INST] When you have expressions which just contains products, quotients and powers, logarithmic differentiation makes life easier.
For your case
$$f = {(x^2+1)(x^2-2) \over 3x+2}\implies \log(f)=\log(x^2+1)+\log(x^2-2)-\log(3x+2)$$ Differentiate both sides
$$\frac{f'}f=\frac{2x}{x^2+1}+\frac{2x}{x^2-2}-\frac 3{3x+2}=\frac{9 x^4+8 x^3-3 x^2-4 x+6 }{(x^2+1 )(x^2-2 ) (3x+2 ) }$$
Now 
$$f'=f \times \frac{f'}f={(x^2+1)(x^2-2) \over 3x+2}\times \frac{9 x^4+8 x^3-3 x^2-4 x+6 }{(x^2+1 )(x^2-2 ) (3x+2 ) }$$ Now, simplify.
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