How do you find the derivative of #1/(x^3-x^2)#?

Answer 1
One law of exponents states that #a^-n=1/a^n#
Thus, we can rewrite this whole expression as #f(x)=(x^3-x^2)^-1#
Now, using the chain rule, we cna rename #u=x^3-x^2#, thus making #f(x)=u^-1#

The chain rule states that

#(dy)/(dx)=(dy)/(du)(du)/(dx)#

So,

#(dy)/(du)=-1*u^-2#
#(du)/(dx)=3x^2-2x#
#(dy)/(dx)=-u^-2(3x^2-2x)#
Substituting #u#:
#(dy)/(dx)=-(x^3-x^2)^-2(3x^2-2x)=-(3x^2-2x)/(x^3-x^2)^2#=#(3x^2-2x)/(x^6-2x^5+x^4)=(cancelx(3x-2))/(cancelx(x^5-2x^4+x^3))#

Thus, final answer:

#(3x-2)/(x^5-2x^4+x^3)#
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Answer 2

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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Answer from HIX Tutor

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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