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

Answer 1

#(1) : dy/dx=1/(xlnx)#.

#(2) : (d^2y)/dx^2=-(1+lnx)/(x^2(lnx)^2)#.

Let #y=ln(lnx^2)#

Using the well-known Rules of Log. Fun., we have,

#y=ln(2lnx)=ln2+ln(lnx)#

By the Chain Rule, then,

#dy/dx=d/dx(ln2)+d/dx{ln(lnx)}#
#=0+1/lnx*d/dx(lnx)#
#:. dy/dx=1/lnx*1/x=1/(xlnx)#.
Before proceeding further for the #2^(nd)# Derivative #(d^2y)/dx^2#, let us recall that #d/dt(1/t)=-1/t^2#. Hence,
#(d^2y)/dx^2=d/dx(dy/dx)=d/dx{1/(xlnx)}#
#=-1/((xlnx)^2)*d/dx{xlnx}#
#=-1/(x^2(lnx)^2){x*d/dx(lnx)+(lnx)*d/dx(x)}#
#=-1/(x^2(lnx)^2){x*1/x+(lnx)(1)}#
#:. (d^2y)/dx^2=-(1+lnx)/(x^2(lnx)^2)#.

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Answer 2

To find the first and second derivatives of ( \ln(\ln(x^2)) ):

  1. First Derivative: ( \frac{d}{dx}(\ln(\ln(x^2))) = \frac{1}{\ln(x^2)} \cdot \frac{d}{dx}(\ln(x^2)) ) Apply chain rule and derivative of natural logarithm: ( = \frac{1}{\ln(x^2)} \cdot \frac{1}{x^2} \cdot 2x )

  2. Second Derivative: ( \frac{d^2}{dx^2}(\ln(\ln(x^2))) = \frac{d}{dx} \left( \frac{1}{\ln(x^2)} \cdot \frac{1}{x^2} \cdot 2x \right) ) Apply quotient rule and derivative of ( \frac{1}{\ln(x^2)} ): ( = -\frac{2}{x^3\ln(x^2)} + \frac{1}{x^4} \cdot 2x \cdot \left( -\frac{1}{x^2} \right) \cdot \frac{d}{dx}(\ln(x^2)) ) ( = -\frac{2}{x^3\ln(x^2)} - \frac{2}{x^3} )

Therefore, the first derivative is ( \frac{2}{x(\ln(x))^2} ) and the second derivative is ( -\frac{2}{x^3(\ln(x))^2} - \frac{2}{x^3} ).

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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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