How do you find the derivative of #f(x)=1/x^2# using the limit process?
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To find the derivative of ( f(x) = \frac{1}{x^2} ) using the limit process, you can follow these steps:

Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x+h)  f(x)}{h} ]

Substitute the function ( f(x) = \frac{1}{x^2} ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2}  \frac{1}{x^2}}{h} ]

Simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{x^2  (x+h)^2}{h(x^2)(x+h)^2} ]

Expand ( (x+h)^2 ) and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{x^2  (x^2 + 2hx + h^2)}{h(x^2)(x^2 + 2hx + h^2)} ] [ f'(x) = \lim_{h \to 0} \frac{x^2  x^2  2hx  h^2}{h(x^2)(x^2 + 2hx + h^2)} ] [ f'(x) = \lim_{h \to 0} \frac{2hx  h^2}{h(x^2)(x^2 + 2hx + h^2)} ] [ f'(x) = \lim_{h \to 0} \frac{2x  h}{(x^2)(x^2 + 2hx + h^2)} ]

Cancel out the common factor of ( h ) in the numerator and denominator: [ f'(x) = \lim_{h \to 0} \frac{2x  h}{(x^2)(x^2 + 2hx + h^2)} ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = \frac{2x}{x^4} ] [ f'(x) = \frac{2}{x^3} ]
So, the derivative of ( f(x) = \frac{1}{x^2} ) with respect to ( x ) is ( f'(x) = \frac{2}{x^3} ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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