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The implicit differentiation that we learned and used in lesson 3.6 is here used to find derivatives of logarithmic functions. We first note that logarithmic functions appear to be differentiable, because their graphs appear to be continuous, with no cusp and no vertical tangent lines. See the graph of the logarithmic functions below:
The derivative of a logarithm:
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Let
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This is our given basic logarithmic function |
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This is the exponential form of the logarithmic function listed above. |
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We use implicit differentiation on both sides of the equation. Note that on the left side, we use our new derivative of a constant to a variable power from the last lesson. |
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Now we solve for  . |
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We now use from step 2 to make a substituion and finish the
proof of this new derivative. |
We also have this special result:
Using the general derivative of a logarithm (above), we would
answer 
. But since 
we know that 
Another
important derivative:
If 
, this
could also be written as: 
Then 
. And
we can see that the above derivative is valid for all values of x
except 0.
Derivatives
of Logarithmic Functions
With various complex combinations of products, quotients, etc., derivatives can get burdensome. Sometimes it is to your advantage to first take the logarithm of the item to be differentiated prior to differentiating, and then differentiate implicitly.
Look at the following example: Use logarithmic differentiation
ot find the derivative of 
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First we take the natural logarithm of both sides |
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Now we differentiate both sides of the equation. |
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Now we solve for y, substituting the orginal value of y in terms of x as defined in the function. |
Here is
a summary of this logarithmic differentiation technique:
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