Examples

Derivative of vector function, constant basis

Derivative of vector function, constant basis

Consider $\mathbf{r}(t)=\ln (t){\mathbf{e}}_{\mathbf{i}}+t{\mathbf{e}}_{\mathbf{j}}+t^3{\mathbf{e}}_{\mathbf{k}}$. Find ${\mathbf{r}}^{\prime }(t)$ and ${\mathbf{r}}^{\prime \prime }(t)$.

Solution: Differentiation is given componentwise, so we can treat the basis vectors as constants and differentiate their coefficients: ${\mathbf{r}}^{\prime }(t)=\frac{1}{t}{\mathbf{e}}_{\mathbf{i}}+{\mathbf{e}}_{\mathbf{j}}+3t^2{\mathbf{e}}_{\mathbf{k}}$ and ${\mathbf{r}}^{\prime \prime }(t)=-\frac{1}{t^2}{\mathbf{e}}_{\mathbf{i}}+6t{\mathbf{e}}_{\mathbf{k}}$.