Question

Let k be a scalar and $r\left(t\right)$be a differentiable vector function. Prove that $\frac{d}{dt}\left(kr\right(t\left)\right)=kr\text{'}\left(t\right)$. (This is Theorem 11.11 (a).)

Short answer

Ans: $\frac{d}{dt}(k⟨x(t),y(t),z(t)⟩)=k{r}^{\text{'}}\left(t\right)$

Step-by-step solution

Step 1. Given information:

k is scalar and

$r\left(t\right)$be a differentiable vector function

Step 2. Proving:

Consider,

Note: Let $\mathrm{u}$be a function of $x$whose derivates exist.

Then $\frac{d}{dx}\left(ku\right)=k\frac{du}{dx}$