Question

Derive
F(x)= x^2/x-1

Answer 1

u = x²
v = x - 1

u' = 2x
v' = 1

u'v = 2x*(x - 1) = 2x² - 2x
uv' = x²(1) = x²

u'v - uv' = 2x² - 2x - x² = x² - 2x
v² = (x-1)² = x² - 2x + 1

Answer 2

Regra do quociente:

$\frac{d}{dx} ( \frac{u}{v} ) = \frac{u'v - uv'}{ v^{2} }$

$\frac{d}{dx} ( \frac{ x^{2} }{x-1} ) = ?$

u = x²
v = x - 1

u' = 2x
v' = 1

u'v = 2x*(x - 1) = 2x² - 2x
uv' = x²(1) = x²

u'v - uv' = 2x² - 2x - x² = x² - 2x
v² = (x-1)² = x² - 2x + 1


Substituindo na regra...

$\frac{d}{dx} ( \frac{ x^{2} }{x-1} ) = \frac{ x^{2} - 2x}{ x^{2} - 2x + 1}$