Use the fundamental definition of a derivative to find f′(x) where f(x)=x+ax+b . Show all your work. If limit notation is needed in your response, use the following example as a guide: limx→1(x2+2) should be written as lim x→1 (x2+2) .

Answer 1

The fundamental definition of a derivative states that the derivative of a function f(x) at a specific point x is given by the limit of the difference quotient as h approaches 0:

f′(x) = limh→0 (f(x+h) - f(x))/h

Now let's find f′(x) for the function f(x) = x + ax + b:

f′(x) = limh→0 (f(x+h) - f(x))/h
= limh→0 ((x+h) + a(x+h) + b - (x + ax + b))/h
= limh→0 (x + h + ax + ah + b - x - ax - b)/h
= limh→0 (h + ah)/h
= limh→0 (h(1 + a))/h
= limh→0 (1 + a)
= 1 + a

Therefore, f′(x) = 1 + a.