Find the Difference quotient:

4/(x-3)^2

4/(x+h-3)^2 - 4/(x-3)^2

---------------------------
h

-4(h+2(x-3))
------------------------------
(x-3)^2(x+h-3)^2

Not sure where else you want to take it.

We have to use the equation

f(x+Δ)-f(x)
-----------
Δx

well, that's what I did, using h for Δx.

To find the difference quotient, we need to calculate the slope of the secant line between two points on the given function and then take the limit as the two points get closer and closer.

The difference quotient for the function f(x) = 4/(x-3)^2 is represented by the expression:

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

where h represents the "difference" or the gap between the two points.

Let's proceed step by step:

1. Start with the function f(x) = 4/(x-3)^2.

2. Substitute x + h into the function to find f(x + h):
f(x + h) = 4/((x + h) - 3)^2

3. Calculate f(x + h) - f(x):
f(x + h) - f(x) = [4/((x + h) - 3)^2] - [4/(x - 3)^2]

4. Combine the fractions by finding a common denominator:
f(x + h) - f(x) = [4(x - 3)^2 - 4((x + h) - 3)^2] / [(x - 3)^2((x + h) - 3)^2]

5. Simplify the expression:
f(x + h) - f(x) = [4(x^2 - 6x + 9) - 4(x^2 + 2hx + h^2 - 6x - 6h + 9)] / [(x^2 - 6x + 9)(x^2 + 2hx + h^2 - 6x - 6h + 9)]

6. Expand and cancel out like terms:
f(x + h) - f(x) = [4x^2 - 24x + 36 - 4x^2 - 8hx - 4h^2 + 24x + 24h - 36] / [(x^2 - 6x + 9)(x^2 + 2hx + h^2 - 6x - 6h + 9)]

7. Combine like terms:
f(x + h) - f(x) = [-8hx - 4h^2 + 24h] / [(x^2 - 6x + 9)(x^2 + 2hx + h^2 - 6x - 6h + 9)]

8. Divide each term by h to obtain the difference quotient:
f(x + h) - f(x) / h = [-8x - 4h + 24] / [(x^2 - 6x + 9)(x^2 + 2hx + h^2 - 6x - 6h + 9)]

9. Finally, take the limit as h approaches 0 to find the slope of the tangent line at a specific point.

That's how you find the difference quotient for the function 4/(x-3)^2.