Logarithmic differentiation

Definition

Logarithmic differentiation is a method used to differentiate functions by taking the natural logarithm of both sides, simplifying using logarithmic properties, and then differentiating implicitly. It is particularly useful for functions that are products or quotients of powers.

5 Must Know Facts For Your Next Test

1. Logarithmic differentiation can simplify the process of differentiating complex products and quotients.
2. It involves taking the natural logarithm ($\ln$) of both sides of an equation before differentiating.
3. Use the property $\ln(ab) = \ln(a) + \ln(b)$ to break down the function into simpler terms.
4. After applying logarithms, use implicit differentiation to find the derivative.
5. Remember to multiply by the original function at the end if solving for $y'$ in terms of $y$.

Review Questions

• What are the steps involved in logarithmic differentiation?
• Why is logarithmic differentiation particularly useful for functions that involve products or quotients?
• How do you apply implicit differentiation after taking the natural logarithm?

Related terms

Implicit Differentiation: A technique to find the derivative of a function defined implicitly rather than explicitly.

$\ln$ (Natural Logarithm): The natural logarithm is the logarithm to the base $e$, where $e \approx 2.71828$.

Exponential Function:

A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.