Logarithmic Differentiation

The problem this technique solves. None of the differentiation rules developed so far

directly handle a function of the form $y=[f(x)]^{g(x)}$, where both the base $f(x)$ and the

exponent $g(x)$ are themselves functions of $x$ -- for instance, $y=x^x$. The power rule

$d(x^n)/dx=nx^{n-1}$ requires a constant exponent $n$; the exponential rule

$d(a^x)/dx=a^x\ln a$ requires a constant base $a$. In $y=x^x$, neither is constant, so

neither rule applies as it stands. Logarithmic differentiation resolves exactly this

situation, by using the property $\ln(f^g)=g\ln f$ (Section 6) to move the variable exponent down

into an ordinary product, which the chain and product rules can then handle.

Method (variable base and exponent). Given $y=[f(x)]^{g(x)}$ with $f(x)>0$:

1. Take the natural logarithm of both sides: $\ln y = g(x)\,\ln\big(f(x)\big)$.
2. Differentiate both sides with respect to $x$, using implicit differentiation on the left (since $y$ is a function of $x$: $d(\ln y)/dx = (1/y)\,dy/dx$, by the chain rule of Section 3) and the product rule on the right:

3. Multiply both sides by $y=[f(x)]^{g(x)}$ to isolate $dy/dx$.

Worked illustration ($y=x^x$, $x>0$). Here $f(x)=g(x)=x$. Taking logs: $\ln y=x\ln x$.

Differentiating (product rule on the right, since both factors depend on $x$):

Multiplying back by $y=x^x$:

matching Example 7.

A second, equally important use: products and quotients of several factors. Even when the

exponents are all constant, logarithmic differentiation turns a lengthy product or quotient --

which would otherwise need repeated applications of the product and quotient rules, with a high

chance of a sign or term-dropping slip -- into a sum of simple logarithmic terms, each

differentiated independently. For $y=\dfrac{u_1u_2\cdots u_m}{v_1v_2\cdots v_n}$ (each $u_i,v_j$

a function of $x$), taking logs gives

so, differentiating term by term,

When to reach for logarithmic differentiation. Two clear signals: (i) the function has a

variable raised to a variable power (like $x^x$, $x^{\sin x}$, $(\sin x)^x$), where no ordinary …