3.3: Chain rule
- Page ID
- 6490
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For a function \(f(x, y)\) and a curve \(\gamma (t) = (x(t), y(t))\) the chain rule gives
\[\dfrac{df(\gamma (t))}{dt} = \left. \dfrac{\partial f}{\partial x} \right\vert_{\gamma (t)} x'(t) + \left. \dfrac{\partial f}{\partial y} \right \vert_{\gamma (t)} y'(t) = \nabla f(\gamma (t)) \cdot y'(t) \text{ dot product of vectors.} \nonumber \]
Here \(\nabla f\) is the gradient of \(f\) defined in the next section.