List of mathematical formulas: “Logarithmic derivative

The logarithmic derivative is a useful tool in mathematics, especially in the analysis of functions. Here are the basic formulas associated with the logarithmic derivative:

1. Definition of logarithmic derivative:
   [
   \frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}
   ]
   where ( f(x) ) is the differentiable function.
2. Logarithmic derivative for a product:
   If ( f(x) = g(x) \cdot h(x) ), then:
   [
   \frac{d}{dx}(\ln(f(x))) = \frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}
   ]
3. Logarithmic derivative for a quotient:
   If ( f(x) = \frac{g(x)}{h(x)} ), then:
   [
   \frac{d}{dx}(\ln(f(x))) = \frac{g'(x)}{g(x)} – \frac{h'(x)}{h(x)}
   ]
4. Logarithmic derivative for a power:
   If ( f(x) = g(x)^{h(x)} ), then:
   [
   \frac{d}{dx}(\ln(f(x))) = h(x) \cdot \frac{g'(x)}{g(x)} + g(x) \cdot \ln(g(x)) \cdot h'(x)
   ]
5. Applications of the logarithmic derivative:
   The logarithmic derivative is often used to simplify calculations of derivatives of complex functions, especially when they are presented as products or quotients.