Chapter 3 - Derivatives - 3.9 Derivatives of Logarithmic and Exponential Functions - 3.9 Exercises - Page 210: 1

\[\frac{dy}{dx}=\frac{1}{x}\]

\[\begin{align} & \text{Let }x={{e}^{y}} \\ & \text{Differentiate both sides with respect to }x \\ & \frac{d}{dx}\left[ x \right]=\frac{d}{dx}\left[ {{e}^{y}} \right] \\ & 1={{e}^{y}}\frac{dy}{dx} \\ & \text{Solve for }\frac{dy}{dx} \\ & \frac{1}{{{e}^{y}}}=\frac{dy}{dx} \\ & or \\ & \frac{dy}{dx}=\frac{1}{{{e}^{y}}} \\ & \text{Back-substitute }x\text{ for }{{e}^{y}} \\ & \frac{dy}{dx}=\frac{1}{x} \\ \end{align}\]