function help

Question

Find the derivative of the function below.

 

f(x)=x^2 (x+4)−2x/(x^2+1)

Answer 1

Hi Daniel, 

 

There are a few different ways to approach this problem.

 

I would simplify the numerator a bit and then use the quotient rule.

 

y = (x³+4x²-2x)/(x²-1)

 

y' = denominator x derivative of numerator - numerator x derivative of denominator all over the denominator squared (at least that is how I repeat the quotient rule to myself (but not usually outloud).

 

Whoops, I see the -2x is the numerator of the fraction, so ignore the above.

 

y = (x³-4x²) - 2x/(x²-1)

 

y' = (3x²-8x) - quotient rule.

 

 

Answer 2

So x^2*(x+4) = x^3 + 4x^2 The derivative of this part is 3x^2 + 8x.

 

As for the 2x/(x^2+1) part, we do lowdihigh - highdilow over lowlow or (2(x^2 +1) - 4x^2)/(x^4 + 2x^2 + 1)

 

So the derivative of the whole function is:

 

3x^2 + 8x -  (2(x^2 +1) - 4x^2)/(x^4 + 2x^2 + 1)