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The Derivative of a Logarithm. The Derivative of a Logarithm. If f ( x ) = log a x , then. Notice if a = e , then. The Derivative of a Logarithm. If f ( x ) = log a g ( x ) and g ( x ) is differentiable, then. Notice if a = e , then. Examples.

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The derivative of a logarithm1
The Derivative of a Logarithm

If f(x) = logax, then

Notice if a = e, then


The derivative of a logarithm2
The Derivative of a Logarithm

If f(x) = logag(x) and g(x) is differentiable, then

Notice if a = e, then


Examples
Examples

Find the derivative of each function:


Example 2
Example 2

Find the derivative of:

You CAN use Logarithm Laws to expand to simplify finding the derivative.

Now take the derivative


Example 3
Example 3

Find the derivative of:

Rewrite as a piece-wise function.

Now take the derivative of each piece

Equal


Example 3 generalized
Example 3 (Generalized)

Find the derivative of:

Rewrite as a piece-wise function.

Now take the derivative of each piece

Equal

The derivative of each piece will always be equal


The derivative of a logarithm composed with an absolute value
The Derivative of a Logarithm Composed with an Absolute Value

If f(x) = loga│g(x)│and g(x) is differentiable, then

Notice if a = e, then

Ignore the Absolute Value.


White board challenge
White Board Challenge Value

Is the function below differentiable at x = 0?


Example 4
Example 4 Value

If it is very complicated or impossible…

Find the derivative of:

Take the natural log of both sides to expand the complicated quotient/product.

Now take the derivative of both sides.

Solve for f '


Logarithmic differentiation
Logarithmic Differentiation Value

  • Take the natural logarithm of both sides.

  • Simplify the “x” side using the properties of logarithms.

  • Differentiate both sides of the equation.

  • Solve for y'.


Example 5
Example Value5

x is in the base and exponent, so power and exponential rules do not apply.

Find the derivative of:

Take the natural log of both sides.

Now take the derivative of both sides.

Solve for y '


1982 ab free response 5
1982 ValueAB Free Response 5


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