Q. 32 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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,

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