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Mth 272
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Derivative of Natural Log
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The derivative of the natural log of x is 1/x
y= ln(x)
y`= 1/x
Therefore if we had a function y= ln(x+2)
y`= 1/(x+2)
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If we are given the function y= ln(x+2)^2 by the chain rule:
y`= (1/(x+2)^2) * 1 = (x+2)^-2 = -2(x+2)^-3 * 1 = -(2/(x+2)^3)
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This function is a composite f(g(x)) where g(x) = (x+2)^2 and f(z) = ln(z).
The derivative is g ' (x) * f ' (g ( x) ).
g ' (x) = 2 ( x+2 ), so the result is
2 ( x + 2) * (1 / (x+2)^2 ) = 2 / (x + 2).
Alternatively, by the properties of logs
ln ((x+2)^2) = 2 ln ( x + 2 ).
The derivative of ln(x + 2) is
(x + 2) ' * 1 / (x + 2) = 1 / (x + 2),
so the derivative of 2 ln(x + 2) is 2 / (x + 2), in agreement with the previous result.
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This is how i would evaluate it, however, while using a solving utility on my computer i noticed a few peculiarities.
y=ln(x+2)^2 -it attempts to solve it as the composite of the 3 functions, f(x)= ln(x+2)^2 = 2ln(x+2)
g(x)= ln(x+2) = (1/(x+2))
h(x)= (x+2) = 1
y`=(2ln((x+2)))/(x+2)
If i input the notation as y=ln((x+2)^2) it solves it as the composite of the 3 functions:
f(x)= 2ln(x+2) = 2/x+2
g(x)= x+2 = 1
y`= 2/x+2
If i was looking at a picture of the problem it would appear as though we were taking the natural log of (x+2)^2, NOT taking the natural log of x+2 raised to the second power but either could technically be true because there's no parenthesis. For the first quotation i believe my method would be correct, for the second quotation, the solvers.
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After a rigorous amount of time on this I think i see the problem, after applying the chain rule to y=ln(x+2)^2 I should have got
y`= 1/(x+2)^2 * 1 or just 1/(x+2)^2
and not gone any further. If i took the second derivative then i would get
y`` = (x+2)^-2 = -2(x+2)^-3
What is the proper way to take the derivative of ln(x+2)^2?
Is it acceptable to leave it in the form 1/(x+2)^2?
If i see on a test the function f(x)= natural log of something raised to a power, f(x) is the composite of which functions?
For such a problem as displayed on a test, which of the two solvers notation above is actually correct?
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Very good question.
However you still don't have the right result.
You basically jumped a step ahead of yourself and then failed to get out of the resulting bind. A common phenomenon shared by the best of minds.
Check my note for two ways to get the result. One involves a very careful and systematic application of the chain rule, another relies mostly on a basic property of logs.
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