Let $f: M \to M$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian manifold, and let $\mu$ be an $f$-invariant probability measure for which the Lyapunov exponents are defined. Ruelle's inequality gives \begin{align*} h_\mu(f) \le \int_M \sum_{\lambda_i(x)>0} \lambda_i(x)\,m_i(x)\,d\mu(x), \end{align*} where $m_i(x)$ is the multiplicity of the Lyapunov exponent $\lambda_i(x)$. Equality holds under the additional hypotheses of Pesin's entropy formula, for example when $\mu$ is absolutely continuous with respect to Riemannian volume in the classical smooth setting, or more generally for suitable SRB measures with absolutely continuous conditional measures on unstable manifolds.