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Dated: 30-10-2024

More about Euler's Theorem Chain Rule

In general, the order of the nth partial derivative1 can be changed without affecting the final result whenever the function2 and all of its partial derivative of order \(\le n\) are continuous.3
If the first 3 partial derivatives1 are continuous3 then

\[f_{xyy} = f_{yxy} = f_{yyx}\]

Or in another notation,

\[ \frac{\partial^3 f}{\partial y^2 \partial x} = \frac{\partial^3 f}{\partial y \partial x \partial y} = \frac{\partial^3 f}{\partial x \partial y^2} \]

The Chain Rule4

If \(y\) is a function2 of \(w\) which is a function2 of \(v\) which is a function2 of \(x\) ultimately then

\[\frac{dy}{dx} = \frac{dy}{dw} \cdot \frac{dw}{dv} \cdot \frac{dv}{dx}\]

Now imagine we have \(w = f(x, y)\) and \(x = g(t)\) and \(y = f(t)\)
then

\[\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt}\]

Pasted image 20241007232746.png

Example

\[w = xy, x = \cos t, \text{ and } y = \sin t\]

$$ \frac{\partial w}{\partial y} = x \quad \text{and} \quad \frac{\partial w}{\partial x} = y $$

$$ \frac{dx}{dt} = -\sin t, \quad \frac{dy}{dt} = \cos t $$

$$ \frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} $$

$$ = (\sin t)(-\sin t) + (\cos t)(\cos t) $$

$$ = -\sin^2 t + \cos^2 t = \cos 2t $$

References

Read more about notations and symbols.

  1. Read more about partial derivatives. ↩↩

  2. Read more about functions. ↩↩↩↩

  3. Read more about continuity. ↩↩

  4. Read more about the chain rule. ↩