Applying the definiton of derivative, find f'(x) if f(x)=7/x b. Find f'(6)

The definition of the derivative of a function is `lim_(Delta x ->0)(f(x+Delta x)-f(x))/(Delta x)` ;assuming the limit exists.

Thus if `f(x)=7/x` then :

`f'(x)=lim_(Delta x -> 0)(7/(x+Delta x)-7/x)/(Delta x)`

`=lim_(Delta x -> 0)((7x-7x-7Delta x)/(x(x+Delta x)))/(Delta x)`

`=lim_(Delta x -> 0)((-7 Delta x)/(x(x+Delta x)))/(Delta x)`

`=lim_(Delta x -> 0)(-7)/(x(x+Delta x))`

`=(-7)/x^2`

Thus `f'(x)=-7/x^2`

Then `f'(6)=-7/(6^2)=-7/36`

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An alternative to find the derivative at a point c for a function differentiable at c is:

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

Here c=6 and f(6)=7/6 so:

`f'(6)=lim_(x->6)(7/x-7/6)/(x-6)`

`=lim_(x->6)((42-7x)/(6x))/(x-6)`

`=lim_(x->6)((-7(x-6))/(6x))/(x-6)`

`=lim_(x->6)-7/(6x)`

`=-7/36` as above.

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