How do you find the first and second derivative of #ln(x^3)#?

Answer 1

#d/dxln(x^3)=3/x#

#(d^2)/(dx^2)ln(x^3)=-3/x^2#

Using the chain rule, the power rule, and the product rule, along with the derivative #d/dx ln(x) = 1/x#, we have

First Derivative:

#d/dxln(x^3) = 1/x^3(d/dxx^3)#
#=1/x^3(3x^2)#
#=3/x#

Second Derivative:

#(d^2)/(dx^2)ln(x^3) = d/dx(d/dxln(x^3))#
#=d/dx(3/x)#
#=d/dx3x^(-1)#
#=3(-x^-2)#
#=-3/x^2#
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Answer 2

To find the first and second derivatives of ( \ln(x^3) ):

  1. First Derivative: [ \frac{d}{dx}\left(\ln(x^3)\right) = \frac{1}{x^3} \cdot \frac{d}{dx}(x^3) ]

  2. Second Derivative: [ \frac{d^2}{dx^2}\left(\ln(x^3)\right) = \frac{d}{dx}\left(\frac{1}{x^3}\right) ]

Then, simplify the expressions as needed.

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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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