Using the limit definition, how do you find the derivative of #f(x) = x/(x+4)#?

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Answer 1 · 2016-07-03T08:52:32

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

#f(x) = x/(x+4)#

#f'(x)=lim_{h to 0} (f(x+h) - f(x))/(h)#

# =lim_{h to 0} 1/(h) ((x+h)/(x + h +4) - x/(x+4))#

# =lim_{h to 0} 1/(h) ( (x+h)(x+4) - x(x+h+4))/ ((x+h+4)(x+4))#

# =lim_{h to 0} 1/(h) ( (x^2 + 4x + hx + 4h - x^2- xh-4x))/ ((x+h+4)(x+4))#

# =lim_{h to 0} 1/(h) ( 4h )/ ((x+h+4)(x+4))#

# =lim_{h to 0} ( 4 )/ ((x+h+4)(x+4))#

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

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