Derivative of Operated Vector-Valued Functions

Topics: Derivative of a Vector-Valued Function - Vector-Valued Function - Calculus

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(theorem)

Suppose that $r(t)$ and $s(t)$ are differentiable vector-valued functions, that $f(t)$ is a differentiable scalar function (i.e. $\mathbb{R}\to \mathbb{R}$), and that $c$ is any scalar constant.

Then, the derivative of the sum:

$$\frac{d}{dt}[r(t)+s(t)]=r^{\prime }(t)+s^{\prime }(t)$$

The derivative of the scalar multiplication:

$$\frac{d}{dt}[cr(t)]=c{r}^{\prime }(t)$$

$$\frac{d}{dt}[f(t)r(t)]=f^{\prime }(t)r(t)+f(t)r^{\prime }(t)$$

The derivative of the inner product (dot product):

$$\frac{d}{dt}[r(t)\cdot s(t)]=r^{\prime }(t)\cdot s(t)+r(t)\cdot s^{\prime }(t)$$

When $r,s:\mathbb{R}\to {\mathbb{R}}^3$, the derivative of the cross product:

$$\frac{d}{dt}[r(t)\times s(t)]=r^{\prime }(t)\times s(t)+r(t)\times s^{\prime }(t)$$