Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=7x+32`?

Answer 1

`[(deltay)/(deltax)]_(limdeltax->0) = (dy)/(dx)=7xx1 = 7`

Explanation:

I prefer to use `delta` instead of `Delta`

Given:`" "f(x)=7x+32`

Write as: `y=7x+32` .....................Equation(1)

Let `x` progress by the small amount of `deltax`
Thus `y` will change by the small amount `deltay`

So after the progression Equation(1) becomes

`y+deltay=7(x+deltax)+32`

`y+deltay=7x+7deltax+32`......................Equation(2)

Subtract Equation(1) from EQuation(2)

`y+deltay=7x+7deltax+32`
`ul(y" "=7x" "+32) larr" Subtract"`
`" "deltay =color(white)(.)0+7deltax+0`

Divide both sides by `deltax`

`(deltay)/(deltax) = (7deltax)/(deltax)`

`[(deltay)/(deltax)]_(limdeltax->0) = (dy)/(dx)=7xx1 = 7`