Logarithmic Differentiation is a technique used in calculus to differentiate functions that are complicated in the form of products, quotients, or powers where both the base and the exponent involve variables. It involves taking the natural logarithm of both sides of the equation before differentiating. This method simplifies the differentiation process, especially when dealing with functions that are hard to differentiate directly.

Steps for Logarithmic Differentiation:

1. Take the natural logarithm of both sides of the given function.

   • If $y = f(x)$, then take the natural logarithm of both sides: $\ln(y) = \ln(f(x))$

2. Use logarithmic properties to simplify the equation.

   • Apply the logarithmic rules, such as: $\ln(a^b) = b \ln(a) \quad \text{and} \quad \ln(ab) = \ln(a) + \ln(b)$
   • For example, if the function is $y = (g(x))^{h(x)}$, taking the natural log gives: $\ln(y) = h(x) \ln(g(x))$

3. Differentiate both sides with respect to $x$, applying the chain rule to the left-hand side where $y$ is a function of $x$:

   • Since $\frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx}$, you will differentiate the logarithmic expression.
   • Differentiate the right-hand side using the chain rule and product rule if necessary.

4. Solve for $\frac{dy}{dx}$ after differentiating.

Example:

Suppose you need to differentiate $y = x^x$. Instead of applying direct differentiation (which is difficult), you can use logarithmic differentiation:

1. Take the natural logarithm of both sides: $\ln(y) = \ln(x^x)$ Using the power rule for logarithms: $\ln(y) = x \ln(x)$

2. Differentiate both sides with respect to $x$:

   • Left side: $\frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx}$
   • Right side: $\frac{d}{dx}[x \ln(x)] = \ln(x) + 1$ Therefore: $\frac{1}{y} \frac{dy}{dx} = \ln(x) + 1$

3. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = y(\ln(x) + 1)$ Since $y = x^x$, we substitute: $\frac{dy}{dx} = x^x (\ln(x) + 1)$

Summary of when to use logarithmic differentiation:

• When the function involves products, quotients, or powers with variable exponents or bases.
• When differentiating functions where both the base and the exponent are variable and it becomes difficult to apply standard differentiation rules.

Would you like more detailed examples or have any specific questions about this technique?

Follow-up Questions:

1. How do you apply logarithmic differentiation to functions involving products of multiple terms?
2. Can logarithmic differentiation be used for implicit differentiation? How?
3. What are some common mistakes when using logarithmic differentiation?
4. How can logarithmic differentiation help in finding derivatives of functions involving trigonometric identities?
5. How do you apply logarithmic differentiation to rational functions (quotients)?

Tip: Always remember that logarithmic differentiation is especially useful when exponents or products become too complicated to handle with basic differentiation rules.