Capital $$(k_t,t\ge 0)$$ in the Solow growth model solves a nonlinear ODE defined by $$b:[0,\infty )\rightarrow \mathbb {R}$$ where $$b^{\prime }(k)\rightarrow \infty $$ as $$k\rightarrow 0^{+}$$ . Macroeconomics states Solow’s ODE has a unique positive equilibrium point $$k^{*}$$ which is asymptotically stable. Proving $$k_t\rightarrow k^{*}$$ as $$t\rightarrow \infty $$ when $$k_0>0$$ is close to $$k^{*}$$ traditionally rests on a Taylor expansion of $$b(k_t)$$ around $$k^{*}$$ and approximation of $$|k_t-k^{*}|$$ . However, this proof ignores the remainder $$R(k_t,k^{*})$$ in Taylor’s formula. Failure to establish $$R(k_t,k^{*})\rightarrow 0$$ as $$t\rightarrow \infty $$ means the accepted proof is incorrect. Our paper presents a correct proof of asymptotic stability in Solow’s model, driven by properties of $$(k_t, t\ge 0)$$ not linearized stability theory, which is rendered useless by explosion of $$b^{\prime }$$ . Derivative explosion appears in many investment models of economic dynamics making our method of general interest.