Wie unterscheidet man (x ^ 2 + x + 3) / sqrt (x-3) anhand der Quotientenregel?

Antworten:

#h '(x) = - 3 (x + 1) / ((x-3) ^ (3/2)) #

Erläuterung:

Die Quotientenregel; gegeben #f (x)! = 0 #

ob #h (x) = f (x) / g (x) #; dann #h '(x) = g (x) * f' (x) - f (x) * g '(x) / (g (x)) ^ 2 #

gegeben #h (x) = (x ^ 2 + x + 3) / root () (x-3) #

Lassen #f (x) = x ^ 2 + x + 3 #

#Farbe (rot) (f '(x) = 2x + 1) #

Lassen #g (x) = Wurzel () (x-3) = (x-3) ^ (1/2) #

#Farbe (blau) (g '(x) = 1/2 (x-3) ^ (1 / 2-1) = 1/2 (x-3) ^ (- 1/2) #

#h '(x) = (x-3) ^ (1/2) * Farbe (rot) ((2x + 1)) - Farbe (blau) (1/2 (x-3) ^ (- 1 / 2)) (x ^ 2 + x + 3) / (Wurzel () (x-3) ^ 2 #

Den größten gemeinsamen Faktor ausrechnen # 1/2 (x-3) ^ (- 1/2) #

#h '(x) = 1/2 (x-3) ^ (- 1/2) (x-3) (2x + 1) - (x ^ 2 + x + 3) / (x-3) #

# => h '(x) = 1/2 (x ^ 2 + x-6x-3-x ^ 2-x-3) / (x-3) ^ (3/2) #

#h '(x) = (-6x-6) / (2 (x-3) ^ (3/2)) #

#h '(x) = - 6 (x + 1) / (2 (x-3) ^ (3/2)) #

#Farbe (rot) (h '(x) = - 3 (x + 1) / ((x-3) ^ (3/2))) # Antworten