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.