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

Answer 1

#dy/dx=(3-lnx)/(3x^(4/3))#

#y = lnx/root3x = lnx/x^(1/3)#

Applying the quotient rule:

#dy/dx = (x^(1/3)* 1/x - lnx* 1/3x^(-2/3))/x^(2/3)#
#= (x^(-2/3) - 1/3lnx*x^(-2/3)) / x^(2/3)#
#= (x^(-2/3)(1-lnx/3))/x^(2/3)#
#=(1-lnx/3)/x^(4/3)#
#=(3-lnx)/(3x^(4/3))#
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Answer 2

To find the derivative of (\frac{\ln x}{x^{1/3}}), you can use the quotient rule. The quotient rule states that if you have a function of the form ( \frac{u}{v} ), then the derivative is given by ( \frac{u'v - uv'}{v^2} ), where ( u' ) and ( v' ) represent the derivatives of ( u ) and ( v ) respectively.

Let ( u = \ln x ) and ( v = x^{1/3} ). Then ( u' = \frac{1}{x} ) and ( v' = \frac{1}{3}x^{-2/3} ).

Applying the quotient rule:

[ \frac{d}{dx} \left( \frac{\ln x}{x^{1/3}} \right) = \frac{(1/x)(x^{1/3}) - (\ln x)((1/3)x^{-2/3})}{(x^{1/3})^2} ]

Simplify this expression to get the derivative of the function.

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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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