,

โดยกฏลูกโซ่จะได้

$$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$$

,

$$\begin{eqnarray*} \frac{dy}{dx} & = & \frac{d}{dx}\left[\log_{3}5x+\text{sech}\sqrt{x}\right]\\ & = & \frac{d}{dx}\log_{3}5x+\frac{d}{dx}\text{sech}\sqrt{x}\\ & = & \frac{d}{dx}\frac{\ln5x}{\ln3}-\text{sech}\sqrt{x}\tanh\sqrt{x}\cdot\frac{d}{dx}\sqrt{x}\\ & = & \frac{1}{\ln3}\cdot\frac{1}{5x}\cdot\frac{d}{dx}5x-\text{sech}\sqrt{x}\tanh\sqrt{x}\cdot\frac{1}{2\sqrt{x}}\\ & = & \frac{1}{x\ln3}-\frac{1}{2\sqrt{x}}\text{sech}\sqrt{x}\tanh\sqrt{x} \end{eqnarray*}$$

,

$$\begin{eqnarray*} \frac{dx}{dt} & = & \frac{d}{dt}\left[\frac{3^{\sin t}}{\ln10}\right]\\ & = & \frac{1}{\ln10}\cdot\frac{d}{dt}3^{\sin t}\\ & = & \frac{1}{\ln10}\cdot3^{\sin t}\cdot\ln3\cdot\frac{d}{dt}\sin t\\ & = & \frac{\ln3}{\ln10}\cdot3^{\sin t}\cdot\cos t\\ & = & \left(\log3\right)3^{\sin t}\cos t \end{eqnarray*}$$

,

$$\begin{eqnarray*} \frac{dy}{dt} & = & \frac{dy}{dx}\cdot\frac{dx}{dt}\\ & = & \left[\frac{1}{x\ln3}-\frac{1}{2\sqrt{x}}\text{sech}\sqrt{x}\tanh\sqrt{x}\right]\cdot\left[\left(\log3\right)3^{\sin t}\cos t\right] \end{eqnarray*}$$