Logarithmic Differentiation

Imagine you were faced with a function like this:

We've never dealt with a variable in the exponent! How can we solve for the derivative when we're faced with this?

Whenever you're faced with a function with a variable in the exponent, utilize logarithmic differentiation to take the derivative!

How to solve with logarithmic differentiation

The procedure for logarithmic differentiation is the following:

  1. Take the natural log (ln) of both side of your function
  2. Simplify your expression as much as possible
  3. Go through implicit differentiation steps

Take the natural log (ln) of both side of your function

This step is pretty easy: simply surround both sides of the equation with a natural log (ln)!

Simplify your expression as much as possible

For this step, we're going to rely on the following log property:

How does this apply to our equation?

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