Anonymous

# FINDING SLOPE USING DEFINITION OF DERIVATIVE (f(x+h) - f(x))/h?

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

x / (x-2) at point (3,3)

Relevance

I'm warning you to begin with that it will look quite messy, but I'll try my best. For f(x+h) just plug in the binomial (x+h) wherever there is an x in the original equation.

{ [ (x+h) / (x+h-2) ] - [ x / (x-2) ] } / h

Simplify the top fraction using the LCD (x+h-2)(x-2)

{ [ (x+h)(x-2) - x(x+h-2) ] / (x+h-2)(x-2) } / h

Simplify the top and condense the bottom.

[ x² + xh - 2x - 2h - x² - xh + 2x ] / h(x+h-2)(x-2)

The initial function should always cancel out, if it doesnt you know you've done something wrong.

[ -2h ] / h(x+h-2)(x-2)

h's cancel.

-2 / (x+h+2)(x-2)

take the limit has h -> 0

-2 / (x+0-2)(x-2)

-2 / (x-2)(x-2)

-2 / (x-2)²

Now plug in (3,3)

-2 / (3-2)²

-2 / 1²

-2

There is your slope, now take that and the point to make the equation of the line

y = -2(x-3) + 3

y = -2x + 6 + 3

y = -2x + 9