Find the derivative using ...?

Question

the alternative form of the derivative ---
the limit as x approaches c of ( f (x) - f (c) ) / x - c.

the problem is f (x) = 4 - (x-3)^2 and c=2.

Showing work is very much appreciated.

Answer 1

First, write the limit:

[x→2]lim (f(x) - f(2))/(x-2)

Evaluate the function:

[x→2]lim ((4 - (x-3)²) - (4 - (2-3)²))/(x-2)

Simplify:

[x→2]lim (4 - (x-3)² - 4 + (-1)²)/(x-2)
[x→2]lim (1 - (x-3)²)/(x-2)

Expand:

[x→2]lim (1 - (x²-6x+9))/(x-2)

Simplify and factor:

[x→2]lim (1 - x²+6x-9)/(x-2)
[x→2]lim -(x²-6x+8)/(x-2)
[x→2]lim -(x-2)(x-4)/(x-2)

Cancel, and evaluate the limit:

[x→2]lim -(x-4)
-(2-4)
2

Answer 2

ok so:

1. 4-(x-3)^2-(4-(2-3)^2)/X-2

for the f(c) part you just plug in 2 because c is 2.

2. so just simplify from there it would get messy if i did it here, someone else might post it, but I got to run sorry.