Use the definition of the derivative to find f'(x) if f(x)=2/x+1

Answer 1

f(x) = 2/(x+1)

f(x+h) = 2/(x+h+1)

f(x+h) - f(x) = 2/(x+h+1) - 2/(x+1)

put over a common denominator

=(2x +2 - 2x-2h-2)/(x+h+1)(x+1)

= -2h/(x+h+1)(x+1)

divide by h

= -2/(x+h+1)(x+1)

take the limit as h approaches zero, replace h with 0

= -2/(x+1)^2

check the answer by using a short cut rule for derivatives of a quotient or product

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

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

but maybe you meant the problem to read

f(x) = 2/x +1 = (2/x)+1

then

f(x+h) = 2/(x+h) + 1

f(x+h) -f(x) = 2/(x+h) +1 - 2/x -1

= 2/(x+h)-2/x

put over a common denominator

(2x-2x-2h)/(x(x+h))

= -2h/(x^2 +xh)

divide by h

= -2h/(x^2+xh)

let h=0

= -2/x^2

check the answer

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

f'(x) = -2x^-2 = -2/x^2