Question

Obtain from the first principle y=1/x^2

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the derivative of y = 1/x^2 using first principles, we will start by writing down the definition of the derivative:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h

Given that f(x) = 1/x^2, we can substitute into the formula to obtain:

f'(x) = lim(h->0) [(1/(x+h)^2 - 1/x^2) / h]

Next, we will simplify the expression:

f'(x) = lim(h->0) [(1/(x^2 + 2hx + h^2) - 1/x^2) / h]
f'(x) = lim(h->0) [(x^2 - x^2 - 2hx - h^2) / (x^2(x^2 + 2hx + h^2)) / h]
f'(x) = lim(h->0) [(-2hx - h^2) / (x^2(x^2 + 2hx + h^2)) / h]
f'(x) = lim(h->0) [-2hx - h^2 / h(x^2)(x^2 + 2hx + h^2)]
f'(x) = lim(h->0) [-2x - h / x^2(x^2 + 2hx + h^2)]
f'(x) = [-2x - 0 / x^2(x^2)]
f'(x) = -2/x^3

Therefore, the derivative of y = 1/x^2 using first principles is:

y' = -2/x^3