Logarithmic Differentiation

Really large quotient. Expressions that have x in the base and x in the exponent

Example

Find general derivative: $y=x^x$ We rearrange $\ln y=\ln x^x$ $\ln y=x\ln x$ Derivative of both sides $\frac{1}{y}\frac{dy}{dx}=\ln x+x\frac{1}{x}$ Multiply both sides b $y$ $\frac{dy}{dx}=y(\ln x+1)$ $\frac{dy}{dx}=x^x(\ln x+1)$

Use Cases of Logarithmic Differentiation

Quotient Rule Replacement

$h(x)=\frac{(5x-3{)}^2}{(3x+2{)}^4(2x+1{)}^3}$ $\ln$ both sides and use logarithm rules $\ln h(x)=\ln (5x-3{)}^2-\ln (3x+2{)}^4-\ln (2x+1{)}^3$ More log rules $\ln h(x)=2\ln (5x-3)-4\ln (3x+2)-3\ln (2x+1)$ Now derive $\frac{1}{h(x)}{h}^{\prime }(x)=2(\frac{1}{5x-3}\ast 5)-4(\frac{1}{3x+2}\ast 3)-3(\frac{1}{2x+1}\ast 2)$ $h^{\prime }(x)=\frac{(5x-3{)}^2}{(3x+2{)}^4(2x+1{)}^3}\ast (\frac{10}{5x-3}-\frac{12}{3x+2}-\frac{6}{2x+1})$ This works for all quotients