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

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Presentation transcript:

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 Find the derivative of each function:

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

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

Example 3 (Generalized) Find the derivative of: Now take the derivative of each piece Equal The derivative of each piece will always be equal Rewrite as a piece-wise function.

The Derivative of a Logarithm Composed with an Absolute Value If f(x) = log a │g(x)│ and g(x) is differentiable, then Notice if a = e, then Ignore the Absolute Value.

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

Example 4 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 ' If it is very complicated or impossible…

Logarithmic Differentiation 1.Take the natural logarithm of both sides. 2.Simplify the “ x ” side using the properties of logarithms. 3.Differentiate both sides of the equation. 4.Solve for y'.

Example 5 Find the derivative of: Take the natural log of both sides. Now take the derivative of both sides. Solve for y ' x is in the base and exponent, so power and exponential rules do not apply.

1982 AB Free Response 5