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Question
Find the derivative of the following function from first principle.
`1/x^2`
Solution
`f(x) = 1/x^2, f(x + h) = 1/(x + h)^2`
∴ `f(x + h) - f(x) = 1/(x + h)^2 - 1/x^2`
= `(x^2 - (x + h)^2)/(x^2 (x + h)^2)`
= `(x^2 - [x^2 + 2xh + h^2])/(x^2 (x + h)^2)`
= `(-h(2x + h))/(x^2(x + h)^2)`
`f'(x) = lim_(h → 0) (f(x + h) - f(x))/h`
= ` lim_(h → 0) (-h(2x + h))/(x^2 (x + h)^2 h)`
= `(-2x)/(x^2x^2)`
= `-2/x^3`
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