Logarithmic Derivative of a Power Over 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 an expression like this:
?
We can use the product and quotient properties 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
= 1.6.
First plot
y
vs.
x
. Then check your plot by clicking “Next”.
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Determine
y
(1.6).
We get
Add the point at
x
= 1.6 to your diagram. Then check your plot by clicking “Next”.
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Add the tangent line at
x
= 1.6. Then check your plot by clicking “Next”.
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Using
y
(1.6), determine
We get
Is this consistent with the graph?
Yes, the slope in the diagram is positive and less steep 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