📈 Logarithmic Derivative Calculator

Understanding Logarithmic Differentiation

Logarithmic and exponential functions are central to modeling growth, decay, and compound interest. The **Logarithmic Derivative Calculator** specializes in two main tasks: standard differentiation of $\ln(x)$ and $e^x$ functions, and the advanced technique of **logarithmic differentiation** for functions with variable exponents, like $y = x^x$.

Core Logarithmic and Exponential Derivative Rules

The calculation relies on these fundamental rules, often combined with the Chain Rule for composite functions ($g(x)$):

• $$\frac{d}{dx} (\ln(x)) = \frac{1}{x}$$ [Image of derivative of natural log]
• $$\frac{d}{dx} (\log_b(x)) = \frac{1}{x \cdot \ln(b)}$$ (using Change of Base: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$)
• $$\frac{d}{dx} (e^x) = e^x$$
• $$\frac{d}{dx} (b^x) = b^x \cdot \ln(b) \text{ where } b \text{ is a constant}$$

Using Logarithmic Differentiation for $f(x)=g(x)^{h(x)}$

For complex functions where both the base $g(x)$ and the exponent $h(x)$ contain the variable $x$ (e.g., $f(x)=x^{\sin(x)}$), the method of **logarithmic differentiation** must be applied. [Image of Logarithmic Differentiation Steps]

This method involves three key steps:

1. Take the natural logarithm ($\ln$) of both sides of the equation: $\ln(f(x)) = \ln(g(x)^{h(x)})$.
2. Apply logarithm properties to simplify the right side (e.g., $\ln(g(x)^{h(x)}) = h(x) \cdot \ln(g(x))$).
3. Differentiate both sides with respect to $x$ (using implicit differentiation and the Product Rule on the right side).

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