Question

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

Short answer

$\frac{df}{dt}=\frac{v}{\sqrt{u+w}}\left(\frac{du}{dt}+\frac{dw}{dt}\right)+2\sqrt{u+w}\frac{dv}{dt}$

Step-by-step solution

Step 1. Given information

A function,$f\left(t\right)=2v\sqrt{u+w}$

Step 2. Derivative of given function

Differentiating the given function, we get,

$$\begin{gathered} \frac{df}{dt}=\frac{2v}{2\sqrt{u+w}}\left(\frac{du}{dt}+\frac{dw}{dt}\right)+2\sqrt{u+w}\frac{dv}{dt} \\ \Rightarrow \frac{df}{dt}=\frac{v}{\sqrt{u+w}}\left(\frac{du}{dt}+\frac{dw}{dt}\right)+2\sqrt{u+w}\frac{dv}{dt} \end{gathered}$$