Question

Use the denition of the derivative to find f 0 (3), where f (x) = 3x+5 / 2x−1

Answer 1

`f'(3)= -(13)/(25)`

Explanation:

We are asked to find
`f'(3)` of function
`f(x)=(3x+5)/(2x-1)` using definition of derivatives.

Limit definition of derivatives:

`f'(x)= \lim_(h \to 0) (f(x+h)-f(x))/(h)`

Let us find
`f(3+h)` and
`f(3)`.

`f(3+h)=(3(3+h)+5)/(2(3+h)-1)`

`f(3+h)=(9+3h+5)/(6+2h-1)\\\\f(3+h)=(3h+14)/(2h+5)`

`f(3)=(3(3)+5)/(2(3)-1)`

`f(3)=(9+5)/(6-1)\\\\f(3)=(14)/(5)`

Substituting these values in limit definition of derivatives, we will get:

`f'(3)= \lim_(h \to 0) (f(3+h)-f(3))/(h)`

`f'(3)= \lim_(h \to 0) ((3h+14)/(2h+5)-(14)/(5))/(h)`

Make a common denominator:

`f'(3)= \lim_(h \to 0) (((3h+14)*5)/((2h+5)*5)-(14(2h+5))/(5(2h+5)))/(h)`

`f'(3)= \lim_(h \to 0) ((5(3h+14)-14(2h+5))/(5(2h+5)))/(h)`

`f'(3)= \lim_(h \to 0) (5(3h+14)-14(2h+5))/(5h(2h+5))`

`f'(3)= \lim_(h \to 0) (15h+70-28h-70)/(5h(2h+5))`

`f'(3)= \lim_(h \to 0) (-13h)/(5h(2h+5))`

Cancel out h:

`f'(3)= \lim_(h \to 0) (-13)/(5(2h+5))`

`f'(3)= (-13)/(5(2(0)+5))`

`f'(3)= (-13)/(5(5))`

`f'(3)= -(13)/(25)`

Therefore,
`f'(3)= -(13)/(25)`.