X = 3에서 f (x) = sqrt (x ^ 2e ^ x)의 접선의 방정식은 무엇입니까?

대답:

# y = 11.2x-20.2 #

또는

# y = (5e ^ (3/2)) / 2x-2e ^ (3/2) #

# y = e ^ (3/2) ((5x) / 2-2) #

설명:

우리는:

#f (x) = (x ^ 2e ^ x) ^ (1/2) #

(x ^ 2e ^ x) ^ (- 1/2) / 2 * d / dx x ^ 2e ^ x #

(x ^ 2e ^ x) ^ (- 1/2) / 2 * (2xe ^ x + x ^ 2e ^ x) #

(2xe ^ x + x ^ 2e ^ x) (x ^ 2e ^ x) ^ (- 1/2)) / 2 #

(2xe ^ x + x ^ 2e ^ x) / (2 ^ e ^ x) ^ (1/2)) = (2xe ^ x + x ^ 2e ^ x) / (x ^ 2e ^ x)) #

(3 ^ 2 ^ 3) / (2sqrt (3 ^ 2e ^ 3)) = (5e ^ (3/2)) / 2 ~ ~ 11.2 #

# y = mx + c #

#f (3) = sqrt (9e ^ 3) = 3e ^ (3/2) ~~13.4 #

# 13.4 = 11.2 (3) + c #

# c = 13.4-11.2 (3) = - 20.2 #

# y = 11.2x-20.2 #

또는

# y = (5e ^ (3/2)) / 2x-2e ^ (3/2) #

# y = e ^ (3/2) ((5x) / 2-2) #