Hvordan differentierer du y = (x + 7) ^ 10 (x ^ 2 + 2) ^ 7?

Svar:

#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 #

Forklaring:

# Y = (x + 7) ^ 10 (x ^ 2 + 2) ^ 7 # er af formularen:

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

En ligning af denne form er differentieret som denne:

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

#U (x) # og #V (x) # er begge i formularen:

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

En ligning af denne form er differentieret som denne:

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

(d (x + 7) ^ 10)) / (d (x + 7)) = 1 * 10 (x + 7) ^ 9 #

# = 10 (x + 7) ^ 9 #

(d (x ^ 2 + 2) ^ 7)) (d (x ^ 2 + 2)) = 2x * 7 (x ^ 2 + 2) ^ 6 #

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

Derfor:

# 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 #