use the definition of the derivative to find the derivative of f(x)= (9-5x)/(7+7x)

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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We have the function:

`f(x)= (9-5x)/(7+7x) = (1/7) (9-5x)/(1+x)`

(We factored 7 from the deniminator for simpler function)

`` We need to use the definition to find the derivative.

`==gt f'(x)= lim_(h-gt0) (f(x+h)-f(x))/h `

`= (1/7) lim_(h-gt0) ((9-5(x+h))/(1+x+h) - (9-5x)/(1+x))/h `

`= (1/7) lim_(h-gt0) ((9-5x-5h)(1+x) - (9-5x)(1+x+h))/ (h(1+x+h)(1+x))`

`= (1/7) lim_(h->0) (9-5x-5h+9x-5hx-5x^2-9-9x-9h+5x+5x^2+5xh)/(h(1+x)(1+x+h))`

Now, after we reduce similar terms, we get:

`= (1/7)lim_(h-gt0) (-5h-9h)/(h(1+x)(1+x+h)) `

`=(1/7)lim(h-gt0) (-14h)/(h(1+x+h))`

`` Now we will reduce h.

`=(1/7)lim (-14)/((1+x)(1+x+h)) `

`= (1/7) (-14)/((1+x)(1+x)) `

`= -2/(1+x)^2`

Then ,`f'(x)= -2/(1+x)^2`

`` `` 

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