Če f (x) = cos 4 x in g (x) = 2 x, kako ločiti f (g (x)) z verigo?

Odgovor:

# -8sin (8x) #

Pojasnilo:

Pravilo verige je navedeno kot:

#barva (modra) ((f (g (x))) '= f' (g (x)) * g '(x)) #

Najdemo derivat od #f (x) # in #g (x) #

#f (x) = cos (4x) #

#f (x) = cos (u (x)) #

Uporabiti moramo verižno pravilo #f (x) #

To vem # (cos (u (x)) '= u' (x) * (cos '(u (x)) #

Let #u (x) = 4x #

#u '(x) = 4 #

#f '(x) = u' (x) * cos '(u (x)) #

#barva (modra) (f '(x) = 4 * (- sin (4x)) #

#g (x) = 2x #

#barva (modra) (g '(x) = 2) #

Nadomestitev vrednosti na lastnost nad:

#barva (modra) ((f (g (x))) '= f' (g (x)) * g '(x)) #

# (f (g (x))) '= 4 (-sin (4 * (g (x))) * 2 #

# (f (g (x))) '= 4 (-sin (4 * 2x)) * 2 #

# (f (g (x))) '= - 8sin (8x) #

Pokažite, da cos²π / 10 + cos²4π / 10 + cos² 6π / 10 + cos²9π / 10 = 2. Malo sem zmeden, če naredim Cos²4π / 10 = cos² (π-6π / 10) & cos²9π / 10 = cos² (π-π / 10), bo postal negativen kot cos (180 ° - theta) = - costheta v drugi kvadrant. Kako naj dokazujem vprašanje?

Glej spodaj. LHS = cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) + cos ^ 2 ((6pi) / 10) + cos ^ 2 ((9pi) / 10) = cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) + cos ^ 2 (pi- (4pi) / 10) + cos ^ 2 (pi- (pi) / 10) = cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) + cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10) = 2 * [cos ^ 2 (pi / 10) + cos ^ 2 ((4pi) / 10)] = 2 * [cos ^ 2 (pi / 2- (4pi) / 10) + cos ^ 2 ((4pi) / 10)] = 2 * [sin ^ 2 ((4pi) / 10) + cos ^ 2 ((4pi) / 10)] = 2 * 1 = 2 = RHS

Če f (x) = cos5 x in g (x) = e ^ (3 + 4x), kako ločiti f (g (x)) z verigo?

Leibnizova notacija je lahko uporabna. f (x) = cos (5x) Naj bo g (x) = u. Potem izpeljava: (f (g (x))) '= (f (u))' = (df (u)) / dx = (df (u)) / (dx) (du) / (du) = (df (u)) / (du) (du) / (dx) = = (dcos (5u)) / (du) * (d (e ^ (3 + 4x))) / (dx) = = -sin (5u) * (d (5u)) / (du) * e ^ (3 + 4x) (d (3 + 4x)) / (dx) = = -sin (5u) * 5 * e ^ (3 + 4x) ) * 4 = = -20sin (5u) * e ^ (3 + 4x)

Če f (x) = cot2 x in g (x) = e ^ (1 - 4x), kako ločiti f (g (x)) z verigo?

(8e ^ (1-4x)) / sin ^ 2 (2e ^ (1-4x)) ali 8e ^ (1-4x) csc ^ 2 (2e (1-4x)) f (g (x)) = cot2e ^ (1-4x) Naj g (x) = u f '(u) = d / (du) cot2u = d / (du) (cos2u) / (sin2u) = (- 2sin (2u) sin (2u) - 2cos (2u) cos (2u)) / sin ^ 2 (2u) = (- 2sin ^ 2 (2u) -2cos ^ 2 (2u)) / sin ^ 2 (2u) = -2 / sin ^ 2 (2u) g '(x) = - 4e ^ (1-4x) Uporaba verižnega pravila: f' (g (x)) = f '(u) * g' (x) = -2 / sin ^ 2 (2u) * - 4e ^ (1-4x) = -2 / sin ^ 2 (2e ^ (1-4x)) * - 4e ^ (1-4x) = (8e ^ (1-4x)) / sin ^ 2 (2e ^ ( 1-4x)) ali 8e ^ (1-4x) csc ^ 2 (2e (1-4x))