Using the limit definition, how do you differentiate #f(x) =1/(x+2)#?

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Answer 1 · 2016-10-31T23:23:04

#f'(x) = ( -1 ) / ((x+2)^2 ) #

By definition # f'(x) =lim_(hrarr0)( (f(x+h)-f(x))/h ) #

So, with # f(x) = 1/(x+2) # we have:

# f'(x) = lim_(hrarr0)( ( (1/((x+h)+2)) - (1/(x+2)) ) / h ) #

# :. f'(x) = lim_(hrarr0)( ( (1/(x+h+2)) - (1/(x+2)) ) / h ) #

# :. f'(x) = lim_(hrarr0)( ( ((x+2) - (x+h+2) ) / ( (x+h+2)(x+2) )) / h ) #

# :. f'(x) = lim_(hrarr0)( ( x+2 - x-h-2 ) / ( h(x+h+2)(x+2) )) #

# :. f'(x) = lim_(hrarr0)( ( -h ) / ( h(x+h+2)(x+2) )) #

# :. f'(x) = lim_(hrarr0)( ( -1 ) / ((x+h+2)(x+2) )) #

# :. f'(x) = ( -1 ) / ((x+2)(x+2) ) #

# :. f'(x) = ( -1 ) / ((x+2)^2 ) #