Hvordan finder du f '(x) ved hjælp af definitionen af et derivat f (x) = sqrt (x-3)?

Hvordan finder du f '(x) ved hjælp af definitionen af et derivat f (x) = sqrt (x-3)?
Anonim

Svar:

Bare drage fordel af # A ^ 2-b ^ 2 = (a-b) (a + b) #

Svar er:

#F '(x) = 1 / (2sqrt (x-3)) #

Forklaring:

#F (x) = sqrt (x-3) #

#F '(x) = lim_ (h-> 0) (sqrt (x + h-3) -sqrt (x-3)) / h = #

# = Lim_ (h-> 0) ((sqrt (x + h-3) -sqrt (x-3)) * (sqrt (x + h-3) + sqrt (x-3))) / (h (sqrt (x + h-3) + sqrt (x-3))) = #

# = Lim_ (h-> 0) (sqrt (x + h-3) ^ 2-sqrt (x-3) ^ 2) / (h (sqrt (x + h-3) + sqrt (x-3))) = #

# = Lim_ (h-> 0) (x + h-3-x-3) / (h (sqrt (x + h-3) + sqrt (x-3))) = #

# = Lim_ (h-> 0) h / (h (sqrt (x + h-3) + sqrt (x-3))) = #

# = Lim_ (h-> 0) annullere (h) / (annullere (h) (sqrt (x + h-3) + sqrt (x-3))) = #

# = Lim_ (h-> 0) 1 / ((sqrt (x + h-3) + sqrt (x-3))) = #

# = 1 / ((sqrt (x + 0-3) + sqrt (x-3))) = 1 / (sqrt (x-3) + sqrt (x-3)) = #

# = 1 / (2sqrt (x-3)) #