Hvordan differentierer du y = ln ((x-1) / (x ^ 2 + 1))?

Svar:

# Dy / dx = (- x ^ 2 + 2x + 1) / ((x ^ 2 + 1) (x-1)) #

Forklaring:

# Y = ln ((x-1) / (x ^ 2 + 1)) #

# Y = ln (x-1) -ln (x ^ 2 + 1) #

Brug kvoteregler for logaritmer

Differentier nu

# dy / dx = 1 / (x-1) -1 / (x ^ 2 + 1) * d / dx (x ^ 2 + 1)Brug kæderegel

# Dy / dx = 1 / (x-1) -1 / (x ^ 2 + 1) * 2x #

# dy / dx = 1 / (x-1) - (2x) / (x ^ 2 + 1) # Tag lcd'et som ((x-1) (x ^ 2 + 1)

(xx2) 1 (x-1)) - ((2x) (x-1)) / ((x ^ 2 + 1) x-1)))

# Dy / dx = (x ^ 2 + 1-2x ^ 2 + 2x) / ((x ^ 2 + 1) (x-1) #

# Dy / dx = (- x ^ 2 + 2x + 1) / ((x ^ 2 + 1) (x-1)) #