How do you find the derivative of #y= ln ((x^2 (x+1))/ (x+2)^3)#?

Answer 1

# dy/dx=(5x+4)/{x(x+1)(x+2)}#

#y=ln{(x^2(x+1))/(x+2)^3}#
Using the Rules of #Log# function, we have,
#y=lnx^2+ln(x+1)-ln(x+2)^3#
#=2lnx+ln(x+1)-3ln(x+2)#

Diff.ing the L.H.S., we will use the Chain Rule.

E.g., #d/dx{ln(x+1)}=1/(x+1)d/dx(x+1)=1/(x+1)#
#:. dy/dx=2(1/x)+1/(x+1)-3/(x+2)#
#={2(x+1)(x+2)+x(x+2)-3x(x+1)}/{x(x+1)(x+2)}#
#:. dy/dx=(5x+4)/{x(x+1)(x+2)}#
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Answer 2

To find the derivative of ( y = \ln \left( \frac{x^2 (x+1)}{(x+2)^3} \right) ), you would use the chain rule and the rules of logarithmic differentiation. The derivative is:

[ y' = \frac{d}{dx} \ln \left( \frac{x^2 (x+1)}{(x+2)^3} \right) = \frac{1}{\frac{x^2 (x+1)}{(x+2)^3}} \cdot \frac{d}{dx} \left( \frac{x^2 (x+1)}{(x+2)^3} \right) ]

To differentiate ( \frac{x^2 (x+1)}{(x+2)^3} ), you can use the quotient rule. After finding the derivative, substitute it into the equation above to get the final answer.

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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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