Question

Let f be a function of two variables that has continuous partial derivatives and consider the points a(7, 3), b(12, 3), c(7, 7), and d(15, 9). the directional derivative of f at a in the direction of the vector ab is 5 and the directional derivative at a in the direction of ac is 4. find the directional derivative of f at a in the direction of the vector ad. (round your answer to two decimal places.)

Answer 1

The directional derivative of a function $f(x,y)$ in the direction of $\mathbf v$ is given by

$\nabla f(x,y)\cdot\mathbf v$

We have $\vec{ab}=\mathbf b-\mathbf a=(12-7,3-3)=(5,0)$, so that $\|\vec{ab}\|=5$, at which point we're given

$\nabla f(7,3)\cdot\dfrac{(5,0)}5=5\implies1\cdot\dfrac{\partial f}{\partial x}(7,3)+0\cdot\dfrac{\partial f}{\partial y}(7,3)=5$

$\implies\dfrac{\partial f}{\partial x}(7,3)=5$

We're also given that, in the direction of $\vec{ac}=\mathbf c-\mathbf a=(7-7,7-3)=(0,4)$ with $\|\vec{ac}\|=4$, we have

$\nabla f(7,3)\cdot\dfrac{(0,4)}4=4\implies0\cdot\dfrac{\partial f}{\partial x}(7,3)+1\cdot\dfrac{\partial f}{\partial y}(7,3)=4$

$\implies\dfrac{\partial f}{\partial y}(7,3)=4$

So in the direction of $\vec{ad}=\mathbf d-\mathbf a=(15-7,9-3)=(8,6)$, with $\|\vec{ad}\|=10$, we have

$\nabla f(7,3)\cdot\dfrac{(8,6)}{10}=\dfrac1{10}(4,4)\cdot(8,6)=9.60$