Hvordan differentierer du f (x) = x ^ 3sqrt (x-2) sinx ved hjælp af produktreglen?

Svar:

#F '(x) = 3x ^ 2sqrt (x-2) sinx + (x ^ 3sinx) / (2sqrt (x-2)) x ^ 3sqrt (x-2) cosx #

Forklaring:

Hvis #F (x) = g (x) h (x) j (x) #, derefter #F '(x) = g' (x) h (x) j (x) + g (x) h '(x) j (x) + g (x) h (x) j' (x) #

#g (x) = x ^ 3 #

#g '(x) = 3x ^ 2 #

#h (x) = sqrt (x-2) = (x-2) ^ (1/2) #

#h '(x) = 1/2 * (x-2) ^ (- 1/2) * d / dx x-2 #

#COLOR (hvid) (h '(x)) = (x-2) ^ (- 1/2) / 2 * 1 #

#COLOR (hvid) (h '(x)) = (x-2) ^ (- 1/2) / 2 #

#COLOR (hvid) (h '(x)) = 1 / (2sqrt (x-2)) #

#J (x) = sinx #

#J '(x) = cosx #

(x-2) sinx + x ^ 3 1 / (2sqrt (x-2)) sinx + x ^ 3sqrt (x-2) cosx #

#F '(x) = 3x ^ 2sqrt (x-2) sinx + (x ^ 3sinx) / (2sqrt (x-2)) x ^ 3sqrt (x-2) cosx #