How do you find the derivative of #f(x)=x^3+x^2# using the limit process?
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To find the derivative of ( f(x) = x^3 + x^2 ) using the limit process:

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

Substitute the function ( f(x) = x^3 + x^2 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^3 + (x + h)^2  (x^3 + x^2)}{h} ]

Expand the terms: [ f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) + (x^2 + 2xh + h^2)  (x^3 + x^2)}{h} ]

Simplify and cancel out like terms: [ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + 2xh + h^2}{h} ]

Factor out ( h ): [ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2 + 2x + h)}{h} ]

Cancel out ( h ): [ f'(x) = \lim_{h \to 0} 3x^2 + 3xh + h^2 + 2x + h ]

Take the limit as ( h ) approaches 0: [ f'(x) = 3x^2 + 2x ]
So, the derivative of ( f(x) = x^3 + x^2 ) is ( f'(x) = 3x^2 + 2x ).
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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.
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