I was working on the following question about directional derivatives.
The directional derivative of a function $\phi(x, y, z)$ at the point $(2, 0, 3)$ in the direction towards $(3, -2, 3)$ is $1.789$, in the direction towards $(2, 4, 4)$ it is $0.243$, whilst in the direction towards $(4, -1, 2)$ it is zero. By showing that the three first order partial derivatives of $\phi$ with respect to $x,$ $y$ and $z$ are $2.0$, $-1.0$ and $5.0$ respectively, verify that the value of the directional derivative of $\phi$ at $(2, 0, 3)$ in the direction towards $(0, 2, 14)$ is $4.314$.
Now I used the following approach. We know that the directional derivative of a scalar field $\phi$ in the direction of a unit vector $\hat{\boldsymbol{\mathrm{s}}}$ is given by
$$\frac{\partial \phi}{\partial s}=\hat{\boldsymbol{\mathrm{s}}}\cdot\nabla\phi$$
Let the first direction given be $\boldsymbol{\mathrm{s}}_1=(3,-2,3)$. Then $\hat{\boldsymbol{\mathrm{s}}}=\frac{1}{\sqrt {22}}(3,-2,3)$. Substituting into the result above gives
$$\frac{\partial \phi}{\partial s_1}=\frac{1}{\sqrt{22}}\begin{pmatrix}3\\-2\\3\end{pmatrix}\cdot\nabla\phi=\frac{3}{\sqrt{22}}\frac{\partial \phi}{\partial x}-\frac{2}{\sqrt{22}}\frac{\partial \phi}{\partial y}+\frac{3}{\sqrt{22}}\frac{\partial \phi}{\partial z}=1.789$$
However if we substitute the given results $\partial\phi/\partial x=2$, $\partial\phi/\partial y=-1$ and $\partial\phi/\partial z=5$, equality does not hold; nor does it hold if we repeat the procedure for the other two directional derivatives.
Is there a flaw in my approach? Perhaps the results in the question are mistaken? I'd appreciate any feedback.
The procedure is correct, but the vectors that need to be taken are the displacement vectors, i.e. the vector from $(2,0,3)$ to $(3,-2,3)$. In other words, $\hat{\boldsymbol{\mathbb s}}_1=(3,-2,3)-(2,0,3)=(1,-2,0)$.
This gives
$$\frac{\partial \phi}{\partial s_1}=\frac{1}{\sqrt{5}}\begin{pmatrix}1\\-2\\0\end{pmatrix}\cdot\nabla\phi=\frac{1}{\sqrt{5}}\frac{\partial \phi}{\partial x}-\frac{2}{\sqrt{5}}\frac{\partial \phi}{\partial y}$$ which $=1.798$ when substituting the given partial derivatives.