Logarithmic Derivative of a Power Times a Radical
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Find the derivative of
.
How do we proceed?
This problem will be simpler if we take the logarithm first and then take the derivative.
How do we take the log of a product:
?
We can use the product property of logarithms:
How do we take the log of a radical?
We convert the radical to a power.
What power is needed here?
Rewrite the expression for
y
without the radical.
Now apply these ideas to express ln(
y
).
How can we simplify this?
We can use the fact that the log of a quantity raised to a power equals the power times the log of the quantity:
Do that.
Finally, we are ready to take the derivative. What rules of differentiation do we need?
The Chain Rule and the derivative of the natural logarithm:
Set up the differentiation.
Take the indicated derivatives.
We get
.
Solve for the derivative by multiplying each side by y.
.
Are we done?
No, we usually replace y by its equivalent from the original problem.
Do that.
To check this result qualitatively, let’s plot the original function,
y
, for the interval
.
Later, we’ll determine the derivative at
x
= 0.3.
First plot
y
vs.
x
. Then check your plot by clicking “Next”.
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Determine
y
(0.3).
We get
Add the point at
x
= 0.3 to your diagram. Then check your plot by clicking “Next”.
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Add the tangent line at
x
= 0.3. Then check your plot by clicking “Next”.
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Using
y
(0.3), determine
We get
Is this consistent with the graph?
Yes, the slope in the diagram is negative and steeper than a slope of -1.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index