Question

Given that $u=u\left(t\right),v=v\left(t\right),$and $w=w\left(t\right)$are functions of $t$and that $k$is a constant, calculate the derivative $\frac{df}{dt}$of each function $f\left(t\right)$. Your answers may involve $u,v,w,$$\frac{du}{dt},\frac{dv}{dt},\frac{dw}{dt},k$and/or $t.$

$f\left(t\right)=\frac{ut+w}{k}$

Short answer

The derivative is

$\frac{df}{dt}=\frac{t}{k}\frac{du}{dt}+\frac{u}{k}+\frac{1}{k}\frac{dv}{dt}$

Step-by-step solution

Step 1. Given Information

The function is

$f\left(t\right)=\frac{ut+w}{k}$

Step 2. Finding the derivative

The function is

$f\left(t\right)=\frac{ut+w}{k}$

Find derivative with respect to $t$

$\frac{df}{dt}=\frac{d}{dt}\left(\frac{ut+w}{k}\right)$

Take out constant from derivative

$\frac{df}{dt}=\frac{1}{k}\left(\frac{d}{dt}\right(ut)+\frac{d}{dt}(w\left)\right)$

Apply product rule of derivative

$$\begin{gathered} \frac{df}{dt}=\frac{1}{k}(t\frac{du}{dt}+u\frac{d}{dt}(t)+\frac{d}{dt}(w\left)\right) \\ \frac{df}{dt}=\frac{1}{k}(t\frac{du}{dt}+u(1)+\frac{d}{dt}(w\left)\right) \\ \frac{df}{dt}=\frac{t}{k}\frac{du}{dt}+\frac{u}{k}+\frac{1}{k}\frac{dw}{dt} \end{gathered}$$