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

2 Answers
mason m
Nov 16, 2015

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

Explanation:

The Limit Definition of Derivative: #lim_(hrarr0)frac(f(x+h)-f(x))h#
#lim_(hrarr0)frac(-2/(x+h+1)+2/(x+1))h#
#=lim_(hrarr0)frac(-2/(x+h+1)+2/(x+1))(h)*((x+h+1)(x+1))/((x+h+1)(x+1))#
#=lim_(hrarr0)frac(-2(x+1)+2(x+h+1))h#
#=lim_(hrarr0)frac(cancel(-2x)cancel(-2)cancel(+2x)+2hcancel(+2))(h(x+h+1)(x+1))#
#=lim_(hrarr0)frac(2)((x+h+1)(x+1))#

Plug in #0# for #h#.

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

Babette
Nov 16, 2015

The derivative is #2/(x+1)^2#

Explanation:

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

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

#lim_(hrarr0)1/h[(2(x+h+1)-2(x+1))/((x+1)(x+h+1))]#

#lim_(hrarr0)1/h[(2x+2h+2-2x-2)/((x+1)(x+h+1))]#

#lim_(hrarr0)1/h[(2h)/((x+1)(x+h+1))]=2/((x+1)(x+h+1))#

#lim_(hrarr0)2/((x+1)(x+0+1))=2/((x+1)(x+1))=2/(x+1)^2#

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