= Derivative along a curve
{title2=$\frac d{dt}F(\gamma(t))=\nabla F(\gamma(t))\cdot\gamma'(t)$}
If $F:\mathbb R^n\to\mathbb R$ is <differentiable> at the point $\gamma(t)$ of a <differentiable> <parametric curve>, the <multivariable chain rule> gives
$$
\frac d{dt}F(\gamma(t))=\nabla F(\gamma(t))\cdot\gamma'(t).
$$
The <gradient> supplies the local spatial rates; the <velocity vector> $\gamma'(t)$ supplies the direction and rate of movement. Their <dot product> is the change in the sampled <scalar field> per unit parameter. If the parameter is time, this is a rate per unit time; a <directional derivative> per unit distance instead uses a <unit vector>.
Take $F(x,y)=x^2+2y^2$ and the <unit circle> path $\gamma(t)=(\cos t,\sin t)$. Along the path,
$$
F(\gamma(t))=1+\sin^2t,\qquad
\frac d{dt}F(\gamma(t))=2\sin t\cos t=\sin2t.
$$
At $t=\pi/4$, the <gradient> is $(\sqrt2,2\sqrt2)$, the <velocity vector> is $(-1/\sqrt2,1/\sqrt2)$, and their <dot product> is $-1+2=1$. The sampled value is $3/2$ and increases at rate one, even though one coordinate is decreasing. This is why a moving observation must combine every coordinate's contribution.
\Image[/chain-rule-along-a-path.png]
{title=The gradient and path velocity combine to give the rate of change along a curve}
{description=Left: <level curves> of $F(x,y)=x^2+2y^2$, the <unit circle> path, and arrows at $t=\pi/4$ for its <velocity vector> and a scaled <gradient>. Right: the sampled <function> $F(\gamma(t))=1+\sin^2t$ and its <tangent line>, of <slope> one at that parameter. The displayed gradient arrow is scaled for readability, without changing its direction.}
For a <level curve>, $F(\gamma(t))$ is constant, so the <derivative along a curve> is zero and its <velocity vector> is <orthogonal> to the <gradient>. Along other paths, the same formula combines a field's geometry with the observer's motion. https://openstax.org/books/calculus-volume-3/pages/4-5-the-chain-rule[OpenStax, multivariable chain rule and parameter-dependent inputs]{ref}.
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