Logarithmic differentiation

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated. These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are \ln(ab) = \ln(a) + \ln(b), \qquad \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b), \qquad \ln(a^n) = n\ln(a). Higher order derivatives Using Faà di Bruno's formula, the n-th order logarithmic derivative is, \frac{d^n}{dx^n} \ln f(x) = \sum_{m_1+2m_2+\cdots+nm_n=n} \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!} \cdot \frac{(-1)^{m_1+\cdots+m_n-1} (m_1 +\cdots + m_n-1)!}{f(x)^{m_1+\cdots+m_n}} \cdot \prod_{j=1}^n \left(\frac{f^{(j)}(x)}{j!}\right)^{m_j}. Using this, the first four derivatives are, \begin{align} \frac{d^2}{dx^2} \ln f(x) &= \frac{f''(x)}{f(x)} - \left(\frac{f'(x)}{f(x)} \right)^2 \\[1ex] \frac{d^3}{dx^3} \ln f(x) &= \frac{f^{(3)}(x)}{f(x)} - 3 \frac{f'(x) f''(x)}{f(x)^2} + 2 \left(\frac{f'(x)}{f(x)} \right)^3 \\[1ex] \frac{d^4}{dx^4} \ln f(x) &= \frac{f^{(4)}(x)}{f(x)} - 4 \frac{f'(x) f^{(3)}(x)}{f(x)^2} - 3 \left(\frac{f''(x)}{f(x)}\right)^2 + 12 \frac{f'(x)^2 f''(x)}{f(x)^3} - 6 \left(\frac{f'(x)}{f(x)} \right)^4 \end{align} ==Applications==

Products A natural logarithm is applied to a product of two functions f(x) = g(x) h(x) to transform the product into a sum \ln(f(x))=\ln(g(x)h(x)) = \ln(g(x)) + \ln(h(x)). Differentiating by applying the chain and the sum rules yields \frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}, and, after rearranging, yields f'(x) = f(x)\times \left\{\frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}\right\} = g(x) h(x) \times \left\{\frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}\right\} = g'(x) h(x) + g(x) h'(x), which is the product rule for derivatives. Quotients A natural logarithm is applied to a quotient of two functions f(x) = \frac{g(x)}{h(x)} to transform the division into a subtraction \ln(f(x)) = \ln\left(\frac{g(x)}{h(x)}\right) = \ln(g(x)) - \ln(h(x)) Differentiating by applying the chain and the sum rules yields \frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)}, and, after rearranging, yields f'(x) = f(x) \times \left\{\frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)}\right\} = \frac{g(x)}{h(x)} \times \left\{\frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)}\right\} = \frac{g'(x) h(x) - g(x) h'(x)}{h(x)^2}, which is the quotient rule for derivatives. Functional exponents For a function of the form f(x) = g(x)^{h(x)} the natural logarithm transforms the exponentiation into a product \ln(f(x)) = \ln\left(g(x)^{h(x)}\right) = h(x) \ln(g(x)) Differentiating by applying the chain and the product rules yields \frac{f'(x)}{f(x)} = h'(x) \ln(g(x)) + h(x) \frac{g'(x)}{g(x)}, and, after rearranging, yields f'(x) = f(x)\times \left\{h'(x) \ln(g(x)) + h(x)\frac{g'(x)}{g(x)}\right\} = g(x)^{h(x)} \times \left\{h'(x) \ln(g(x)) + h(x) \frac{g'(x)}{g(x)}\right\}. The same result can be obtained by rewriting f in terms of exp and applying the chain rule. General case Using capital pi notation, let f(x) = \prod_i (f_i(x))^{\alpha_i(x)} be a finite product of functions with functional exponents. The application of natural logarithms results in (with capital sigma notation) \ln (f(x)) = \sum_i\alpha_i(x) \cdot \ln(f_i(x)), and after differentiation, \frac{f'(x)}{f(x)} = \sum_i \left[\alpha_i'(x)\cdot \ln(f_i(x)) + \alpha_i(x) \cdot \frac{f_i'(x)}{f_i(x)}\right]. Rearrange to get the derivative of the original function, f'(x) = \overbrace{\prod_i (f_i(x))^{\alpha_i(x)}}^{f(x)} \times\overbrace{\sum_i\left\{\alpha_i'(x)\cdot \ln(f_i(x))+\alpha_i(x)\cdot \frac{f_i'(x)}{f_i(x)}\right\}}^{[\ln (f(x))]'}. ==See also==