Using the 4 step method fidn the derivative of F(x) =1/(x^2)

1) I got /(x^2+2xh+h^2)
2)2xh+h^2/(x^2+2xh+h^2)(x^2)
3)2x+h/(x^2+2xh+h^2)(x^2)
4)1/x

I don't understand why I got this problem wrong on my test can you explain the correct way?

The second step is wrong, it should be lim h->0 [F(x+h)-F(x)]/h
So we should have
lim h->0 1/(x^2+2xh+h^2) - 1/x^2 =
lim h->0 [x^2 - (x^2+2xh+h^2)]/[h*(x^2+2xh+h^2)*x^2] =
lim h->0 -(2xh+h^2)/[h*(x^2+2xh+h^2)*x^2] =
lim h->0 -(2x+h)/[(x^2+2xh+h^2)*x^2] =
lim h->0 -(2x+h)/(x^4+2x^3h+x^2h^2) =
-2x/x^4 =
-2/x^3 = F'(x)

To find the derivative of F(x) = 1/(x^2), we can use the 4-step method for finding derivatives. Let's go through each step correctly:

Step 1: Write down the limit expression:
lim h->0 [F(x+h) - F(x)]/h

Step 2: Substitute F(x) into the expression:
lim h->0 [1/(x^2 + 2xh + h^2) - 1/(x^2)]/h

Step 3: Simplify the expression:
To simplify this, we need to find a common denominator for the two fractions in the numerator. The common denominator is (x^2)(x^2 + 2xh + h^2). After getting a common denominator, we can combine the fractions:
lim h->0 [(x^2) - (x^2 + 2xh + h^2)] / [h(x^2)(x^2 + 2xh + h^2)]

Simplifying further:
lim h->0 [-2xh - h^2] / [h(x^2)(x^2 + 2xh + h^2)]

Step 4: Cancel out the h's and take the limit as h approaches 0:
Canceling out the h in the denominator and setting h to 0:
lim h->0 [-2x] / [(x^2)(x^2)] = -2/x^3

Therefore, the correct derivative of F(x) = 1/(x^2) is -2/x^3.