# How do you find f'(x) using the definition of a derivative for #f(x)= 1/(x-3)#?

##### 1 Answer

Please see the explanation section below.

#### Explanation:

Definition of derivative:

So we have

If we try to evaluate the limit by substitution, we get the indeterminate form

We need to rewrite. Our goal is to make the denominator no longer go to

It is probably not clear to a beginning student what might work, so think about what you **could** do. Then see if that helps.

The smart thing to do here is to write the numerator as a single fraction.

# = lim_(hrarr0)(((x-3)-(x+h-3))/((x-3)(x+h-3)))/h#

# = lim_(hrarr0)((-h)/((x-3)(x+h-3)))/h#

If we try substitution, we still get

# = lim_(hrarr0)((-h)/((x-3)(x+h-3)))/(h/1)#

# = lim_(hrarr0)(-h)/((x-3)(x+h-3))*1/h#

# = lim_(hrarr0)(-1)/((x-3)(x+h-3))#

Now the numerator clearly does not approach

# = (-1)/((x-3)(x+0-3)) = (-1)/(x-3)^2#

That is,