Can You Help Me With This Problem And Include All The Work

Find the derivative by the limit process.

f(x) = 1/x²

lim (f(x + h) - f(x))/h =
h-> 0
lim (1/(x + h)^2 - 1/x^2)/h =
h-> 0
lim (x^2/(x^2(x + h) - (x + h)^2/(x^2(x + h)^2))/h =
h-> 0
lim ((x^2 - (x^2 + 2xh + h^2))/(x^4 + 2hx^2 + h^2x^2))/h =
h-> 0
lim (-2xh - h^2)/(h(x^4 + 2hx^2 + h^2x^2)) =
h-> 0
lim (-2x - h)/(x^4 + 2hx^2 + h^2x^2) = -2x/x^4 = -2/x^3
h-> 0

f(x) = 1/x^2
Using the difference equation
lim
h -> 0

(1/(x+h)^2 - 1/x^2) / h
( 1/(x^2 +2hx + h^2) - 1/x^2) / h
[(x^2 - x^2 - 2hx - h^2)/(x^2)(x^2+2xh+h^2)] / h
(-2hx - h^2) / (x^4 + 2x^3h + x^2h^2)(h)
Factor out an h
(-2x - h) / (x^4 + 2x^3h + x^2h^2)
Now take the limit at h = 0
(-2x-0) / (x^4 + 2x^3(0) + x^2(0)^2)
(-2x) / (x^4)
Factor out an x to get
-2 / x^3

Double check it via power rule:
1/x^2
x^-2
Derive now:
-2x^-3
-2/x^3
It works.

f(x) = 1/x²

f(x+h) = 1/(x+h)²

f(x+h) - f(x)
= 1/(x+h)² - 1/x²
= (x² - (x+h)²) / x²(x+h)²
= (x² - x² - 2hx - h²) / x²(x+h)²
= (-2hx - h² ) / x²(x+h)²
= -h(2x + h ) / x²(x+h)²

[ f(x+h) - f(x) ] / h = -(2x+h) / x²(x+h)²

The limit, as h approaches 0, is -2x/x^4 = -2/x³

f(x) = x⁻², so f’(x) = -2x⁻³ a power rule application, one of the simplest of derivatives.