calculus

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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)

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