Hvordan finder du derivatet af sinx / (1 + cosx)?

Svar:

# 1 / (cosx + 1) #

Forklaring:

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

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

Derivatet af #F (x) / g (x) # Brug af Quotient Rule er

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

så i vores tilfælde er det

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

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

# (Farve (blå) (cos ^ 2x) + cosx + farve (blå) (sin ^ 2x)) / (cosx + 1) ^ 2 # #=#

#cancel ((cosx + farve (blå) (1))) / (cosx + 1) ^ annullere (2) # #=#

# 1 / (cosx + 1) #

Svar:

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

Forklaring:

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

# = Tan (x / 2) #.

# "Derfor" d / dx {sinx / (1 + cosx)} #, # = D / dx {tan (x / 2)} #, # = sec ^ 2 (x / 2) * d / dx {x / 2} …… "Kædelegemet" #, # = Sec ^ 2 (x / 2) * 1/2 #, # = 1 / 2sec ^ 2 (x / 2), eller #

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