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.

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