What intervals is f'(x) = (2x(x^2-25)-2x(x^2))/((x^2-25)^2) increasing and decreasing?

Question

zehanz · Answer

If you simplify f', you get:$$f'(x)=\frac{ 2x(x^2-25)-2x*x^2 }{ (x^2-25)^2 }=\frac{ 2x(x^2-25-x^2) }{ (x^2-25)^2 }$$$$f'(x)=-\frac{ 50x }{ (x^2-25)^2 }$$We have to differentiate f'again and solve f''(x) = 0 to see if where f' is increasing and decreasing:$$f''(x)=-\frac{ 50(x^2-25)^2 -50x*2(x^2-25)*2x }{ (x^2-25)^4 }$$Simplify:$$f''(x)=-\frac{ 50(x^2-25)(x^2-25-4x^2) }{ (x^2-25)^4 }$$$$f''(x)=-\frac{ 50(-3x^2-25) }{ (x^2-25)^3 }$$If you try to solve f''(x) = 0 you can see that 

1. -3x²-25 = 0
2. x = 5 or x = -5 also have influence on the sign of f'', so make a sign scheme with the zeroes of th nominator AND the denominator.

zehanz · Answer

In fact, I hope I made a mistake somewhere, because I don't like the nominator ;)
Try to differentiate yourself (you need the Quotient Rule and the Chain Rule)