Hvordan differentierer du (x ^ 2 + x + 3) / sqrt (x-3) ved hjælp af kvotientreglen?

Svar:

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

Forklaring:

Kvotientreglen givet #F (x)! = 0 #

hvis #h (x) = f (x) / g (x) #; derefter #h '(x) = g (x) * f' (x) -f (x) * g '(x) / (g (x)) ^ 2 #

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

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

#farve (rød) (f '(x) = 2x + 1) #

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

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

#h (x) = (x-3) ^ (1/2) * farve (rød) (2x + 1)) - farve (blå) (1/2 (x-3) ^ (- 1 / 2)) (x ^ 2 + x + 3) / (root () (x-3) ^ 2 #

Faktor ud den største fælles faktor # 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 ^ 2xx) / (x-3) ^ (3/2) #

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

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

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