Examples

Derivative with respect to a vector

Derivative with respect to a vector

Problem: Find the derivative of $f(x,y)=x{e}^y$ at the point $(2,-1)$ with respect to the vector $\mathbf{v}=(2,3)$.
Solution: The partials at $(2,-1)$ are given by $f_x=e^y=e^{-1}$ and $f_y=x{e}^y=2e^{-1}$. So we calculate $D_{\mathbf{v}}(f)(2,-1)=e^{-1}\cdot 2+2e^{-1}\cdot 3\approx 2.94$.

Directional derivative

Directional derivative

Problem: Find the ‘directional’ derivative of $f(x,y)=x{e}^y$ at the point $(2,-1)$ in the direction of the vector $\mathbf{v}=(2,3)$.
Solution: The key phrase “in the direction of” suggests that we need the unit vector in the direction of $\mathbf{v}$. The norm of $\mathbf{v}$ is $\sqrt{13}$, so we want $D_{\frac{1}{\sqrt{13}}\mathbf{v}}(f)$. By the linearity rule with $\lambda =\frac{1}{\sqrt{13}}$, we can just multiply the previous result by $\frac{1}{\sqrt{13}}$, obtaining $0.82$.