Kako pronaći derivat sinx / (1 + cosx)?

Odgovor:

# 1 / (cosx + 1) #

Obrazloženje:

#F (x) = sinx / (cosx + 1) #

#F '(X) = (sinx / (cosx + 1)) #

Derivacija od #F (x) / g (x) * koristeći Quotient Rule je

# (F (x) g (x) f (x) g '(x)) / g ^ 2 (x) *

tako je u našem slučaju

#F "(x) = ((sinx) (cosx + 1) -sinx (cosx + 1)) / (1 + cosx) ^ 2 # #=#

# (Cosx (cosx + 1) + sin ^ 2 x) / (1 + cosx) ^ 2 # #=#

# (Boja (plava) (cos ^ 2x) + cosx + boja (plava) (sin ^ 2 x)) / (1 + cosx) ^ 2 # #=#

#cancel ((cosx + boja (plava) (1))) / (1 + cosx) ^ poništavanje (2) # #=#

# 1 / (cosx + 1) #

Odgovor:

# 1 / 2sec ^ 2 (x / 2) ili 1 / (1 + cosx) #.

Obrazloženje:

Imamo, # Sinx / (1 + cosx) #, # = {2sin (x / 2) cos (x / 2)} / {2cos ^ 2 (x / 2)} #,

# = Tan (x / 2) *.

# "Stoga," d / dx {sinx / (1 + cosx)} #, # = D / dx {tan (x / 2)} #, # = sek ^ 2 (x / 2) * d / dx {x / 2} …… "Pravilo lanca" #, # = Sec ^ 2 (x / 2) * 1/2 #, # = 1 / 2sec ^ 2 (x / 2) ili, #

# = 1 / (2cos ^ 2 (x / 2)) *, # = 1 / (1 + cosx) #.