Logarithmic Differentiation

5.5 Logarithmic Differentiation

Why Logarithmic Differentiation?

The standard rules handle $x^n$ (variable base, constant exponent) and $a^x$ (constant base, variable exponent). But for $[u(x)]^{v(x)}$, where both base and exponent are functions of $x$, neither the power rule nor the exponential rule applies directly. Logarithmic differentiation is the systematic method: take the natural log of both sides of $y = f(x)$, simplify with logarithm properties, then differentiate implicitly.

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The General Method

Let $y = f(x) = [u(x)]^{v(x)}$, with $u(x) > 0$ and $v(x)$ differentiable.

Step 1 — take natural logarithm (base $e$) on both sides:

Step 2 — apply the power rule $\log(a^b) = b \log a$:

Step 3 — differentiate both sides w.r.t. $x$ (chain rule on the left, product rule on the right):

using $\frac{d}{dx}\big(\log[u(x)]\big) = \frac{u'(x)}{u(x)}$ by the chain rule.

Step 4 — equate and multiply both sides by $y$:

Step 5 — substitute back $y = [u(x)]^{v(x)}$:

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Special Case: Differentiating $a^x$ (Constant Base)

This is the general formula with $u(x) = a$ (constant, $a > 0$) and $v(x) = x$.

Method 1 (logarithmic differentiation): Let $y = a^x$. Then $\log y = x \log a$, so …