Given the function f(x)=1/x+7 , calculate the following values: (f(a+h)-f(a))/h

Answer 1

f(x) = 1/x + 7

f(a) = 1/a + 7

f(a+h) = 1/(a+h) + 7

f(a+h)-f(a) = 1/(a+h) + 7 - 1/a -7 = 1/(a+h) - 1/a = (a -a-h)/a(a+h) = -h/a(a+h)

[f(a+h)-f(a)]/h = [-h/a(a+h)]/h = -1/a(a+h)

you could check the answer by calculating f'(x) = -1/x^2 and comparing it to 1/a(a+h) when h=0 and a=x

UNLESS you really intended to write

f(x) = 1/(x+7)

then

f(a) = 1/(a+7)

f(a+h) =1/(a+h+7)

f(a+h)-f(a) = 1/(a+h+7) - 1/(a+7) = [a+7 - a-h-7]/(a+7)(a+h+7) = -h/(a+7)(a+h+7)

[f(a+h)-f(a)]/h = -1/(a+7)(a+h+7)

the derivative of f(x) = f'(x) = -1/(x+7)^2 = [f(a+h)-f(a)]/h when h=0 and a=x