Find the difference quotient?

Question

i need to learn how to find the difference quotient. please provide steps and detail if possible thank you. not all have to be answered

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

2)f(x)= 2x-1

3) f(x)= 1/ √x

Answer 1

That involves a non-trivial amount of algebra.  #2 is the simplest, so I'll give you that one.  You can do the others, it's just going to be tedious.

The difference quotient:

[f(x + h) - f(x)]/h

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

f(x) = 2x - 1

So,  f(x + h) - f(x) = 2x + 2h - 1 - (2x - 1) = 2x + 2h - 1 - 2x + 1 = 2h

So,  [f(x + h) - f(x)]/h = 2h/h = 2

It is always the case that you can cancel the h in the denominator if you carry the algebra far enough.  Whether you want to carry it that far in the case of 1 and 3 is up to you and whatever your teacher told you.  Just remember - it is always possible.

Good luck.