Q 33. - Calculus

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 $t$.

$f\left(t\right)=w{(u+t)}^2$

Step-by-Step Solution

Answer

The derivative is 

$\frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+2w(u+t)(\frac{du}{dt}+1)$

Step 1. Given Information

The function is 

$f\left(t\right)=w{(u+t)}^2$

Step 2. Finding the derivative

The function is 

$f\left(t\right)=w{(u+t)}^2$

Differentiate both sides with respect to $t$

$\frac{df}{dt}=\frac{d}{dt}\left(w{(u+t)}^2\right)$

Apply product rule of derivative

$\frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+w\frac{d}{dt}{(u+t)}^2$

Apply chain rule of derivative

$$\begin{gathered} \frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+w2(u+t)\frac{d}{dt}(u+t) \\ \frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+w2(u+t)(\frac{du}{dt}+\frac{d\left(t\right)}{dt}) \\ \frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+w2(u+t)(\frac{du}{dt}+1) \\ \frac{df}{dt}={(u+t)}^2\frac{dw}{dt}+2w(u+t)(\frac{du}{dt}+1) \end{gathered}$$