= Solution
A <directional derivative> measures change per unit distance, so first normalize the direction: $\widehat v=(4,3)/5$. The <gradient> is
$$
\nabla f=-\frac{2(x,y)}{(1+x^2+y^2)^2},\qquad
\nabla f(1,0)=(-1/2,0).
$$
Taking the <dot product> with the <unit vector>,
$$
\boxed{D_{\widehat v}f(1,0)=(-1/2,0)\cdot(4/5,3/5)=-\frac25.}
$$
Using the unnormalized <vector> would instead give $-2$, the change per unit parameter along $(1,0)+tv$; it is not the rate per unit distance requested here.
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