How do you find the derivative of #1/(x^3-x^2)#?
The chain rule states that
So,
Thus, final answer:
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To find the derivative of ( \frac{1}{x^3 - x^2} ), apply the quotient rule, which states that for functions ( u(x) ) and ( v(x) ),
[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} ]
Let ( u(x) = 1 ) and ( v(x) = x^3 - x^2 ). Then, differentiate ( u(x) ) and ( v(x) ) with respect to ( x ) to find ( u'(x) ) and ( v'(x) ).
[ u'(x) = 0 ] [ v'(x) = 3x^2 - 2x ]
Now, apply the quotient rule:
[ \frac{d}{dx} \left( \frac{1}{x^3 - x^2} \right) = \frac{0 \cdot (x^3 - x^2) - 1 \cdot (3x^2 - 2x)}{(x^3 - x^2)^2} ]
Simplify:
[ \frac{-(3x^2 - 2x)}{(x^3 - x^2)^2} ]
Thus, the derivative of ( \frac{1}{x^3 - x^2} ) is ( -\frac{3x^2 - 2x}{(x^3 - x^2)^2} ).
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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