Find the derivative of each of the following functions using the definition of derivative

Question

a) f(x)=ax+b (Assuming that a and b are non zero constants.)

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

Answer 1

The definition of derivative means the limit as h->0 of the difference quotient. And by difference quotient (slope of a secant line) is meant [f(x+h) - f(x)]/h.

So for part b we have:

lim 2(x+h) 2x

h-> 0 ----------- - ------

(x+h+1) (x+1)

------------------------------

h

The difference quotient can be simplified by treating it as a complex fraction and multiplying the numerator and denominator of the "big fraction" by the LCD of the denominators of the "little factions". The LCD is (x+1)(x+h+1).

Let's simplify then take the limit at the end.

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

---------------------------------

h(x+1)(x+h+1)

2x² +2x +2hx +2h -2x²-2hx-2x

--------------------------------------------

h(x+1)(s+h+1)

2h

---------

h(x+1)(x+h+1)

Finally: 2

-------------------

(x+1)(x+h+1)

Now take the limit as h->0 by simply substituting 0 for h to get the derivative using the definition of derivative:

f'(x) = 2

---------

(x+1)²

^{Thank goodness we eventually get to use the quotient rule!}

Answer 2

f(x)=ax+b => f'(x)=a

f(x) = 2x/(x+1) => f'(x) = 2/(x+1)²

The question may mean do the epsilon-delta method on each function...but that seems a bit extreme since you do the epsilon-delta method on general forms to avoid messy algebra on complex forms!

BTW: Recently (after 60+ years of doing calculus) I learned a mnemonic from some students for the quotient rule:

downsie d upsie - upsie d downsie divided by downsie squared

Goodluck!