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

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Answer 1 · 2016-07-07T16:50:41

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

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 $deltax$

$(deltay)/(deltax) = (7deltax)/(deltax)$

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

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