Logarithmic Differentiation

Ordinary differentiation rules — the power rule, the exponential rule — each handle only one specific relationship between the base and the exponent (a fixed power of a variable base, or a fixed base to a variable power). They both fail on a power-exponential function, where both the base and the exponent depend on $x$: the simplest example is $y=x^x$.

Deriving $y=x^x$. Take the natural logarithm of both sides (valid since $x>0$ here): $\log y = x\log x$. Since this is an identity, differentiating the left side must equal differentiating the right side; the left side needs the chain rule (because $y$ is a function of $x$, so $\log y$ is a function of a function) while the right side needs the product rule:

This technique — take logs, differentiate implicitly, solve for $y'$ — is called logarithmic differentiation, and $\dfrac{d}{dx}\big[\log f(x)\big] = \dfrac{f'(x)}{f(x)}$ is called the logarithmic derivative of $f(x)$. Its real payoff is that it turns products, quotients, and powers inside a function into sums, differences, and constant multiples (via $\log(ab)=\log a+\log b$, $\log(a/b)=\log a-\log b$, $\log(a^n)=n\log a$) before differentiating — often drastically simplifying an otherwise unwieldy product/quotient-rule computation, quite apart from power-exponential functions.

Steps in logarithmic differentiation. (1) Take the natural logarithm of both sides of $y=f(x)$ and simplify using the laws of logarithms. (2) Differentiate implicitly with respect to $x$. (3) Solve the resulting equation for $y'$.

Worked illustration — a product-of-powers. For $y=4x^2\sqrt{x^2+4}\cdot\sin^2 x\cdot 2^x$ (a product of several factors, each itself a power or exponential): $\log y = \log4+2\log x+\tfrac12\log(x^2+4)+2\log(\sin x)+x\log2$. Differentiating, $\dfrac{y'}{y} = \dfrac2x+\dfrac{x}{x^2+4}+2\cot x+\log2$, so $y' = y\left[\dfrac2x+\dfrac{x}{x^2+4}+2\cot x+\log2\right]$ — every product/quotient/power in the original expression became an addable term after taking logs.

The four general exponent/base cases this technique (together with the chain rule) covers completely:

1. $\dfrac{d}{dx}\!\left(a^b\right)=0$ when $a,b$ are both constants (a constant to a constant power is itself just a constant).
2. $\dfrac{d}{dx}\!\left(\big[f(x)\big]^b\right) = b\big[f(x)\big]^{b-1}f'(x)$ — a constant power of a variable base (the ordinary power/chain rule).
3. $\dfrac{d}{dx}\!\left(a^{g(x)}\right) = a^{g(x)}(\log a)\,g'(x)$ — a constant base to a variable power (the exponential/chain rule). …