Ako rozlišujete y = (x + 7) ^ 10 (x ^ 2 + 2) ^ 7?

odpoveď:

#y '= (10 (x ^ 2 + 2) + 14x (x + 7)) (x + 7) ^ 9 (x ^ 2 + 2) ^ 6 #

# = (24x ^ 2 + 98x +20) (x + 7) ^ 9 (x ^ 2 + 2) ^ 6 #

vysvetlenie:

# Y = (x + 7) ^ 10 (x ^ 2 + 2) ^ 7 # má formu:

# Y = U (x) V (x) #

Rovnica tohto formulára sa líši takto:

# Y '= U' (x), V (x) + U (x) V '(x) #

#U (x) # a #V (x) # sú obe z formulára:

#U (x) = G (f (x)) #

Rovnica tohto formulára je diferencovaná takto:

#U '(x) = f' (x) g "(f (x)) #

#rarr U '(x) = (d (x + 7)) / (dx) (d ((x + 7) ^ 10)) / (d (x + 7)) = 1 x 10 (x + 7) ^ 9 #

# = 10 (x + 7) ^ 9 #

#rarr V '(x) = (d (x ^ 2 + 2)) / (dx) (d ((x ^ 2 + 2) ^ 7)) / (d (x ^ 2 + 2)) = 2x * 7 (x ^ 2 + 2) ^ 6 #

# = 14x (x ^ 2 + 2) ^ 6 #

Z tohto dôvodu:

# Y '= 10 (x + 7) ^ 9 (x ^ 2 + 2) ^ 7 + 14x (x + 7) ^ 10 (x ^ 2 + 2) ^ 6 #

# = (10 (x ^ 2 + 2) + 14x (x + 7)) (x + 7) ^ 9 (x ^ 2 + 2) ^ 6 #

# = (24x ^ 2 + 98x +20) (x + 7) ^ 9 (x ^ 2 + 2) ^ 6 #