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Dated: 30-10-2024

Examples

Example 1

\[w = \sin (x) + x^2 \text{ and } x = 3r + 4s\]

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\[\frac{dw}{dx} = \cos{x} + 2x\]

\(\(\frac{\partial x}{\partial r} = 3, \quad \frac{\partial x}{\partial s} = 4\)\)

\(\(\frac{\partial w}{\partial r} = \frac{dw}{dx} \cdot \frac{\partial x}{\partial r}\)\)

\(\(= (\cos{x} + 2x) \cdot 3\)\)

\(\(= 3\cos{(3r + 4s)} + 6(3r + 4s)\)\)

\(\(= 3\cos{(3r + 4s)} + 18r + 24s\)\)

\(\(\frac{\partial w}{\partial s} = \frac{dw}{dx} \cdot \frac{\partial x}{\partial s}\)\)

\(\(= (\cos{x} + 2x) \cdot 4\)\)

\(\(= 4\cos{x} + 8x\)\)

\(\(= 4\cos{(3r + 4s)} + 8(3r + 4s)\)\)

\(\(= 4\cos{(3r + 4s)} + 24r + 32s\)\)

Example 2

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\[\frac{\partial w}{\partial r} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial r} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial r}\]

\(\(\frac{\partial w}{\partial s} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial s}\)\)

Chain Rule1 For Functions2 of Multiple Variables

Suppose \(w = f(x, y, \ldots, v)\) be a function2 of multiple variables, a finite set3 and \(x, y, \ldots, v\) are themselves functions2 of variables \(p, q, t\) etc, another finite set.3
Then,

\[\frac{\partial w}{\partial p} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial p} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial p} + \cdots + \frac{\partial w}{\partial v} \cdot \frac{\partial v}{\partial p}\]
\[\frac{\partial w}{\partial q} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial q} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial q} + \cdots + \frac{\partial w}{\partial v} \cdot \frac{\partial v}{\partial q}\]
\[\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial t} + \cdots + \frac{\partial w}{\partial v} \cdot \frac{\partial v}{\partial t}\]

References

Read more about notations and symbols.

  1. Read more about chain rule. ↩

  2. Read more about functions. ↩↩↩

  3. Read more about sets. ↩↩