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The Derivative of a Logarithm

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 Logarithm

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  1. The Derivative of a Logarithm

  2. The Derivative of a Logarithm If f(x) = logax, then Notice if a = e, then

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

  4. Examples Find the derivative of each function:

  5. Example 2 Find the derivative of: You CAN use Logarithm Laws to expand to simplify finding the derivative. Now take the derivative

  6. Example 3 Find the derivative of: Rewrite as a piece-wise function. Now take the derivative of each piece Equal

  7. 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

  8. 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.

  9. White Board Challenge Is the function below differentiable at x = 0?

  10. Example 4 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 '

  11. Logarithmic Differentiation • 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'.

  12. Example 5 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 '

  13. 1982 AB Free Response 5

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