# Logarithmic Differentiation

### Logarithmic Differentiation:

• Logarithmic differentiation is a technique used to find derivatives of functions that are products, quotients, or powers, especially when the functions are complex or involve variables in both the base and exponent positions.

### Steps to Perform Logarithmic Differentiation:

1. Take the natural logarithm (ln) of both sides of the function you want to differentiate.

2. Use logarithm properties to simplify the expression:

• Apply properties such as $\mathrm{ln}\left(ab\right)=\mathrm{ln}\left(a\right)+\mathrm{ln}\left(b\right)$ or $\mathrm{ln}\left({a}^{n}\right)=n\mathrm{ln}\left(a\right)$.
3. Differentiate implicitly with respect to the variable using the chain rule when necessary.

4. Solve for the derivative in terms of the original function.

### Example:

Given the function $y={x}^{x}$:

1. Take the natural logarithm of both sides: $\mathrm{ln}\left(y\right)=\mathrm{ln}\left({x}^{x}\right)$

2. Apply logarithm properties: $\mathrm{ln}\left(y\right)=x\mathrm{ln}\left(x\right)$

3. Differentiate implicitly: $\frac{1}{y}\cdot \frac{dy}{dx}=\frac{d}{dx}\left(x\mathrm{ln}\left(x\right)\right)$ Using product rule: $\frac{d}{dx}\left(x\mathrm{ln}\left(x\right)\right)=1\cdot \mathrm{ln}\left(x\right)+x\cdot \frac{1}{x}=\mathrm{ln}\left(x\right)+1$

4. Solve for the derivative: $\frac{dy}{dx}=y\left(\mathrm{ln}\left(x\right)+1\right)$ Substituting back $y={x}^{x}$ : $\frac{dy}{dx}={x}^{x}\left(\mathrm{ln}\left(x\right)+1\right)$

### Tips for Logarithmic Differentiation:

• Complex Functions:

• It's particularly useful for functions that involve products, quotients, or powers where direct differentiation becomes cumbersome.
• Logarithmic Properties:

• Familiarize yourself with logarithm properties to simplify expressions before differentiation.
• Handling Multiple Functions:

• Logarithmic differentiation works well when dealing with multiple functions multiplied, divided, or raised to powers.

### Special Cases:

• Exponential Functions:

• Logarithmic differentiation is often beneficial for functions involving exponential terms like ${a}^{x}$ where $a$ is a constant.
• Trigonometric Functions:

• It can be used to simplify derivatives involving trigonometric functions when they are raised to variable powers or involve complex combinations.

• Useful in resolving indeterminate forms like $0\cdot \mathrm{\infty }$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ when dealing with limits.