Question

$$\begin{gathered} \mathrm{Given}\mathrm{that}u=u\left(t\right),v=v\left(t\right),\mathrm{and}w=w\left(t\right)\mathrm{are}\mathrm{functions}\mathrm{of}t\mathrm{and}\mathrm{that}k\mathrm{is}\mathrm{a}\mathrm{constant}, \\ \mathrm{calculate}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{each}\mathrm{function}f\left(t\right)\mathrm{in}\mathrm{Exercises}27–36.\mathrm{Your}\mathrm{answers}\mathrm{may} \\ \mathrm{involve}u,v,w,\frac{du}{dt},\frac{dv}{dt},\frac{dw}{dt},k,and/ort.f\left(t\right)=3u^{2}v+vt \end{gathered}$$

Short answer

$\frac{df}{dt}=6uv\frac{du}{dt}+3u^2\frac{dv}{dt}+t\frac{dv}{dt}+v$

Step-by-step solution

Step 1. Given information

A function,$f\left(t\right)=3u^{2}v+vt$

Step 2. Derivative of given function

Differentiating the given function, we get,

$$\begin{gathered} \frac{df}{dt}=3v\left(2u\right)\frac{du}{dt}+3u^2\frac{dv}{dt}+t\frac{dv}{dt}+v \\ \Rightarrow \frac{df}{dt}=6uv\frac{du}{dt}+3u^2\frac{dv}{dt}+t\frac{dv}{dt}+v \end{gathered}$$