Example 1
\(f(x,y) = xy \to \frac{\partial f}{\partial x} = y, \frac{\partial f}{\partial y} = x\)
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Theory & Formula
Partial derivatives measure how a multivariable function changes with respect to one variable while the others are held constant.
- \(\frac{\partial f}{\partial x}\): Differentiate treating y as constant
- \(\frac{\partial f}{\partial y}\): Differentiate treating x as constant
- Gradient: \(\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\) points toward steepest ascent
- Applications: Optimization, physics, economics, machine learning
\(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\)
Worked Examples
Example 2
\(f(x,y) = x^2y \to \frac{\partial f}{\partial x} = 2xy, \frac{\partial f}{\partial y} = x^2\)
Example 3
\(f(x,y) = x^3y^2 \to \frac{\partial f}{\partial x} = 3x^2y^2, \frac{\partial f}{\partial y} = 2x^3y\)