Hoe onderscheid je (x ^ 2 + x + 3) / sqrt (x-3) met behulp van de quotiëntregel?

Antwoord:

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

Uitleg:

De quotiënt-regel; gegeven #f (x)! = 0 #

als #h (x) = f (x) / g (x) #; dan #h '(x) = g (x) * f' (x) -f (x) * g '(x) / (g (x)) ^ 2 #

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

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

#color (rood) (f '(x) = 2x + 1) #

laat #g (x) = root () (x-3) = (x-3) ^ (1/2) #

#color (blauw) (g '(x) = 1/2 (x-3) ^ (1 / 2-1) = 1/2 (x-3) ^ (- 1/2) #

#h '(x) = (x-3) ^ (1/2) * kleur (rood) ((2x + 1)) - kleur (blauw) (1/2 (x-3) ^ (- 1 / 2)) (x ^ 2 + x + 3) / (root () (x-3) ^ 2 #

Factor uit de grootste gemene deler # 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)) #

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