שאלה # 3dd7c

תשובה:

# = - 2csc2xcot2x #

הסבר:

תן

#f (x) = csc2x #

#f (x + Deltax) = csc2 (x + Deltax) #

#f (x + Deltax) -f (x) = csc2 (x + Deltax) -csc2x #

עכשיו, # (x + Deltax) -deltax) (= (x + Deltax) -csc2x) / (+ Deltax)

# (1 / (Deltax) ((csc2 (x + Deltax) -csc2x) / (Deltax)) #

# 1 / (Deltax) (1 / חטא (2 (x + Deltax)) - 1 / חטא (2x) # #

# (1) (Deltax) ((sin2x-sin2 (x + Deltax)) / (חטא (2 (x + Deltax)) sin2x)) #

# CINC-sinD = 2cos (C + D) / 2) חטא (C-D) / 2) # #

מרמז

# C = 2x, D = 2 (x + Deltax) # #

# (C + D) / 2 = (2x + 2 (x + Deltax)) / 2 #

# = (2x + 2x + 2Deltax) / 2 #

# = (4x + 2Deltax) / 2 #

# = 2 (2x + Deltax) / 2 #

# (C + D) / 2 = 2x + Deltax #

# (C-D) / 2 = (2x-2 (x + Deltax)) / 2 #

# = (2x-2x-2Deltax) / 2 #

# = (- 2Deltax) / 2 #

# (C-D) / 2 = -Deltax #

# sin2x-sin2 (x + Deltax) = 2cos (2x + Deltax) חטא (-Deltax) # #

# (+ דלטקס) - (1) (דלטקס) (2) (2x + Deltax) חטא (דלטקס) (חטא (2 + x Deltax)) sin2x) #

# (2) (- חטא (Deltax) / (Deltax)) (1 / חטא (2x)) (cos (2x + Deltax)) / (חטא (2 (x + Deltax)))

# (2) / sinxlim (Deltaxto0) (חטא (Deltax) / (Deltax)) Lim (Deltaxto0) ((cos (2x + Deltax)) / (חטא (2 + x Deltax))) #)

#lim (Deltaxto0) (חטא (Deltax) / (Deltax)) = 1 #

עכשיו, # = - 2cscx (1) (cos2x) / sin (2x) #

# = - 2csc2xcot2x #