MATH 20C Lecture 8

Differentiable

is differentiable at if

If is differentiable at we denote the above limit by

${\vec{r}}^{\prime }(t)=⟨x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)⟩$

In particular, is differentiable iff all components are differentiable at .

Higher derivatives

Differentiation Rules

Assume that are differentiable.

$({\vec{r}}_1(t)+{\vec{r}}_2(t){)}^{\prime }={\vec{r}}_1^{\prime }(t)+{\vec{r}}_2^{\prime }(t)$

$(c{\vec{r}}_1(t){)}^{\prime }=c{\vec{r}}_1^{\prime }(t)$

Product Rule

Let be a differentiable scalar-valued function

Chain Rule

Let be a differentiable scalar-valued function

Product Rules for the Dot and Cross Products

Dot Product Rule

Cross Product Rule

Geometry of the Derivatives

${\vec{r}}^{\prime }(t_0)$ is a tangent vector to the curve at $t=t_0$

The tangent line at is

Vector-Valued Integration