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

#dy/dx=2(ln(x)+3)/(x^2ln(x)^4)#

We want to find the derivative of

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To find the derivative of ((-2)/(x(\ln x)^3)), you can use the quotient rule of differentiation. The quotient rule states that if you have a function (u(x)) divided by another function (v(x)), then the derivative of (u(x)/v(x)) is given by:

[ \frac{{d}}{{dx}}\left(\frac{{u(x)}}{{v(x)}}\right) = \frac{{v(x) \cdot u'(x) - u(x) \cdot v'(x)}}{{(v(x))^2}} ]

First, let (u(x) = -2) and (v(x) = x(\ln x)^3). Then, find the derivatives (u'(x)) and (v'(x)):

(u'(x) = 0) (since the derivative of a constant is zero)

(v'(x) = \frac{{d}}{{dx}}\left(x(\ln x)^3\right))

Using the product rule, we get:

(v'(x) = x \cdot \frac{{d}}{{dx}}\left((\ln x)^3\right) + (\ln x)^3 \cdot \frac{{d}}{{dx}}(x))

(v'(x) = x \cdot 3(\ln x)^2 \cdot \frac{1}{x} + (\ln x)^3 \cdot 1)

(v'(x) = 3(\ln x)^2 + (\ln x)^3)

Now, plug (u(x)), (v(x)), (u'(x)), and (v'(x)) into the quotient rule formula:

[ \frac{{d}}{{dx}}\left(\frac{{-2}}{{x(\ln x)^3}}\right) = \frac{{x(\ln x)^3 \cdot 0 - (-2) \cdot (3(\ln x)^2 + (\ln x)^3)}}{{(x(\ln x)^3)^2}} ]

[ \frac{{d}}{{dx}}\left(\frac{{-2}}{{x(\ln x)^3}}\right) = \frac{{2 \cdot (3(\ln x)^2 + (\ln x)^3)}}{{(x(\ln x)^3)^2}} ]

[ \frac{{d}}{{dx}}\left(\frac{{-2}}{{x(\ln x)^3}}\right) = \frac{{2(3(\ln x)^2 + (\ln x)^3)}}{{x^2(\ln x)^6}} ]

So, the derivative of ((-2)/(x(\ln x)^3)) is (\frac{{2(3(\ln x)^2 + (\ln x)^3)}}{{x^2(\ln x)^6}}).

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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.

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