`f(x) = |x - 6|, c = 6` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Expert Answers
leonard-chen eNotes educator| Certified Educator

`lim_(x->6) (f(x) - f(c))/(x-c)`

` ` `lim_(x->6) (|x-6|)/(x-6)`

In order to evaluate the above function, both the left hand limit and right hand limit need to be evaluated.

`L.H.L. = lim_(x->6^-) (|x-6|)/(x-6) = lim_(x->6^-) (-(x-6))/(x-6) = -1`

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`R.H.L. = lim_(x->6^+) (|x-6|)/(x-6) = lim_(x->6^+) (x-6)/(x-6) = 1`

``Because the L.H.L. `!=` R.H.L., the limit at x=6 does not exist, and therefore the derivative at x=6 does not exist.

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gsarora17 eNotes educator| Certified Educator


`f'(6)=lim_(h->0) (f(6+h)-f(6))/h`

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`f'(6)=lim_(h->0) ((6+h-6)-(6-6))/h`

`f'(6)=lim_(h->0) h/h`

`f'(6)=lim_(h->0) `

Since the limit is undefined, f'(6) is not defined.