Implicit Differentiation and Logarithmic Derivatives

Yield curves, option pricing models, and capital market line equations define relationships between variables that cannot always be solved explicitly for one in terms of the other. Implicit differentiation treats $y$ as an unknown function of $x$ and differentiates both sides of the relation with respect to $x$, using the chain rule on every occurrence of $y$. The complementary technique — logarithmic differentiation — takes the natural log of both sides before differentiating, turning products into sums and powers into multipliers. Together, these two tools prove the power rule for all real exponents, give a clean formula for $(\ln x)' = 1/x$, and handle $x^x$-type expressions that neither the power rule nor the exponential derivative can reach.

• differentiate implicitly: treat $y$ as a function of $x$, apply the chain rule to each $y$-term, then solve for $dy/dx$;
• find tangent lines and horizontal/vertical tangents for implicitly defined curves;
• derive and apply $\frac{d}{dx}\ln x = 1/x$ and $\frac{d}{dx}\ln|x| = 1/x$;
• use logarithmic differentiation to differentiate products of many factors and variable-exponent expressions like $x^{\sqrt{x}}$;
• state and prove the power rule for all real $n$ using logarithmic differentiation.

1. Implicit differentiation
2. Derivative of $\ln x$
3. Logarithmic differentiation
4. The power rule for all real $n$