Krdytkyu

Do we only calculate directional derivatives for scalar-valued functions?

Is it not possible to calculate directional derivatives for vector-valued functions?

How about using the vector of directional derivatives of the components of the given vector function? Would there be any useful physical or geometric meaning?

For a specific (randomly chosen) example, if $vec v(x,y,z)$ is given by
$$
vec v(x,y,z)=
begin{bmatrix}
x^3+y^2+z\
ze^x\
xyz-9xz\
end{bmatrix}
$$
how can we interpret the directional derivative of $vec v$ at the point $(1,2,3)$ in the direction of the vector $vec u=2i+3j-5k$?

The generic formula for the directional derivative of a function $f$ in the direction $u$ (for a unit vector) is $D_u f (x,y,z) = nabla f(x,y,z) cdot u$. For a vector, just do this to all the components.

Let's look at the example you give. Let's call $f_1(x,y,z) = x^3+y^2+z$, $, f_2(x,y,z) = ze^x$, and $f_3(x,y,z) = xyz - 9xz$. Then the gradients are

$$ begin {align*} nabla f_1 &= (3x^2, , 2y, , 1) \
nabla f_2 &= (ze^x, , 0, , e^x) \
nabla f_3 &= (yz - 9z, , xz, , xy - 9x)
end {align*}
$$

At your particular point $(1,2,3)$, these are:

$$ begin {align*}
nabla f_1(1,2,3) &= (3,4,1) \
nabla f_2(1,2,3) &= (3e,0,e) \
nabla f_3(1,2,3) &= (-27,3,-7)
end {align*}
$$

The formula I mentioned above for directional derivative requires a UNIT vector. Since the vector you give, $u = (2,3,-5)$ is NOT a unit vector, we have to rescale it, and instead use the vector

$$ w = frac{1}{|u|}u = frac{1}{sqrt{38}}(2,3,-5) $$

Now finally use the formula:

$$ begin {align*}
D_w f_1(1,2,3) &= frac{1}{sqrt{38}}(2,3,-5) cdot (3,4,1) = frac{13}{sqrt{38}} \
D_w f_2(1,2,3) &= frac{1}{sqrt{38}}(2,3,-5) cdot (3e,0,e) = frac{e}{sqrt{38}} \
D_w f_3(1,2,3) &= frac{1}{sqrt{38}}(2,3,-5) cdot (-27,3,-7) = frac{98}{sqrt{38}}
end {align*}
$$

Putting it all together, the directional derivative of your vector function $v(x,y,z)$ in the direction u (also the same as direction w) is given by

$$ frac{1}{sqrt{38}} left( begin {array}{c} 13 \ e \ 98 end{array} right) $$