How do you find the first and second derivative of #(lnx)/x^2#?

Answer 1

#d/dxlnx/x^2=(1-2lnx)/x^3# and #d^2/dx^2lnx/x^2=(6lnx-5)/x^4#

Let's rewrite #lnx/x^2# as #x^-2lnx.# If we have a positive exponent in the denominator, we can move the term up to the numerator and in turn make the exponent negative.

It's nice to avoid using the Quotient Rule where unnecessary. It saves time and work.

Using the Product Rule:

#d/dxlnx/x^2=lnx*d/dx(x^-2)+d/dx(lnx)*x^-2#
#d/dxx^-2=-2x^-3# (Power Rule)
#d/dxlnx=1/x#

Thus,

#d/dxx^-2lnx=x^-2(1/x)+(-2x^-3)lnx=x^-2/x-(2lnx)/x^3#

Simplify further:

#x^-2/x-(2lnx)/x^3=x^-3-(2lnx)/x^3=1/x^3-(2lnx)/x^3=(1-2lnx)/x^3#

For the second derivative, we'll use the quotient rule:

#d/dx(1-2lnx)/x^3=(x^3(d/dx1-2lnx)-(1-2lnx)d/dxx^3)/(x^3)^2#
#d/dx(1-2lnx)/x^3=(((-2x^3)/x)-3x^2(1-2lnx))/x^6=(-2x^2-3x^2+6x^2lnx)/x^6=(-5x^2+6x^2lnx)/x^6=(x^2(-5+6lnx))/x^6=(6lnx-5)/x^4#
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Answer 2

To find the first and second derivatives of ( \frac{{\ln x}}{{x^2}} ), you can use the quotient rule and the chain rule.

First Derivative: [ \frac{{d}}{{dx}}\left(\frac{{\ln x}}{{x^2}}\right) = \frac{{x^2 \cdot \frac{{d}}{{dx}}(\ln x) - \ln x \cdot \frac{{d}}{{dx}}(x^2)}}{{(x^2)^2}} ]

Second Derivative: [ \frac{{d^2}}{{dx^2}}\left(\frac{{\ln x}}{{x^2}}\right) = \frac{{d}}{{dx}}\left(\frac{{x^2 \cdot \frac{{d}}{{dx}}(\ln x) - \ln x \cdot \frac{{d}}{{dx}}(x^2)}}{{(x^2)^2}}\right) ]

Simplify the expressions using the derivative of ( \ln x ) and ( x^2 ): [ \frac{{d}}{{dx}}(\ln x) = \frac{{1}}{{x}} ] [ \frac{{d}}{{dx}}(x^2) = 2x ]

Then, substitute these into the first and second derivative expressions and simplify further.

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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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