Home page for accesible maths 3 Chapter 3 contents 3.16 Change in a function along a curve 3.18 Proof of the Chain Rule 1

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3.17 Chain Rule 1

Theorem (Chain rule 1).

Suppose that $C$ is a curve with parametric form $C:(x(t),y(t))$ , and that $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ is a differentiable function. Then the derivative with respect to $t$ of $f(x(t),y(t))$ is

{{df}\over{dt}}={{\partial f}\over{\partial x}}{{dx}\over{dt}}+{{\partial f}% \over{\partial y}}{{dy}\over{dt}}.

Here $f$ can be any function; $f$ is not related to $C$ . As a mnemonic, we can think of the $\partial x$ and the $dx$ ‘cancelling’ in the first term, and the $\partial y$ and the $dy$ ‘cancelling’ in the second term to leave fractions which look like the left hand side. We write $d/dt$ since we are thinking of $f(x(t),y(t))$ as a function of the single variable $t$ . The partial derivatives are computed for $f(x,y)$ .