Logarithmic differentiation is the [[Derivative|differentiation]] of the [[Logarithmic Function|natural logarithm]] of both sides of an equation, then solving the resulting equation for $\frac{dy}{dx}$; used when the function involves complex products. > [!summary] Procedure > > 1. Take the natural logarithms of both sides of an equation $y = f(x)$, and using the laws of logarithms to simplify. > 2. Differentiate [[Implicit Differentiation|implicitly]] with respect to $x$. > 3. Solve the resulting equation for $\frac{dy}{dx}$. > > $ > \begin{align*} > y &= f(x) = g(x) \cdot h(x) \cdot k(x) \\ > \ln{y} &= \ln{(g(x) \cdot h(x) \cdot k(x))} \\ > \ln{y} &= \ln{g(x)} + \ln{h(x)} + \ln{k(x)} \\ > \frac{d}{dx}\ln{y} &= \frac{d}{dx}\ln{g(x)} + \frac{d}{dx}\ln{h(x)} + \frac{d}{dx}\ln{k(x)} \\ > \frac{dy}{dx}\frac{d}{dy}\ln{y} &= \frac{g^\prime{(x)}}{g(x)} + \frac{h^\prime{(x)}}{h(x)} + \frac{k^\prime{(x)}}{k(x)} \\ > \frac{dy}{dx}\frac{1}{y} &= \frac{g^\prime{(x)}}{g(x)} + \frac{h^\prime{(x)}}{h(x)} + \frac{k^\prime{(x)}}{k(x)} \\ > \frac{dy}{dx} &= y\left(\frac{g^\prime{(x)}}{g(x)} + \frac{h^\prime{(x)}}{h(x)} + \frac{k^\prime{(x)}}{k(x)}\right) \\ > \frac{dy}{dx} &= g(x) \cdot h(x) \cdot k(x) \cdot \left(\frac{g^\prime{(x)}}{g(x)} + \frac{h^\prime{(x)}}{h(x)} + \frac{k^\prime{(x)}}{k(x)}\right) > \end{align*} > $