Partial derivatives extend the idea of slope to functions with more than one input. For a surface z = f(x, y), the output can change in different ways depending on whether x changes, y changes, or both change. This matters in physics, engineering, economics, and data science because many quantities depend on several variables at once.
A partial derivative measures one direction of change while the other variables are held constant.
Geometrically, f_x is the slope of the curve you get by slicing the surface parallel to the xz-plane while keeping y fixed. Similarly, f_y is the slope of a slice parallel to the yz-plane while keeping x fixed. At a point, these slopes describe how steeply the surface rises or falls in the x and y directions.
Together, partial derivatives help build tangent planes, gradients, optimization methods, and models of changing systems.
Key Facts
- For z = f(x, y), the partial derivative with respect to x is f_x = ∂f/∂x.
- For z = f(x, y), the partial derivative with respect to y is f_y = ∂f/∂y.
- When finding ∂f/∂x, treat y as a constant.
- When finding ∂f/∂y, treat x as a constant.
- At a point (a, b), f_x(a, b) is the slope of the surface slice where y = b.
- The tangent plane approximation is L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b).
Vocabulary
- Partial derivative
- A derivative of a multivariable function with respect to one variable while all other variables are held constant.
- Multivariable function
- A function whose output depends on two or more input variables, such as z = f(x, y).
- Surface
- The three-dimensional graph of a function z = f(x, y), where height represents the output value.
- Tangent slice
- A two-dimensional curve formed by fixing one input variable and cutting through a surface.
- Tangent plane
- A flat plane that best approximates a smooth surface near a given point.
Common Mistakes to Avoid
- Differentiating every variable at once. A partial derivative changes only one chosen variable and treats the others as constants.
- Treating the fixed variable as zero. Holding y constant in ∂f/∂x means y stays as a constant symbol, not that y = 0.
- Confusing f_x with multiplication by x. The notation f_x means the partial derivative with respect to x, not f times x.
- Forgetting to evaluate at the point after differentiating. To find f_x(a, b), first compute f_x(x, y), then substitute x = a and y = b.
Practice Questions
- 1 For f(x, y) = 3x^2y + 4y^2 - 5x, find ∂f/∂x and ∂f/∂y.
- 2 For f(x, y) = x^2 + xy + y^3, find f_x(2, 1), f_y(2, 1), and the tangent plane approximation at (2, 1).
- 3 A surface z = f(x, y) has f_x(1, 3) = 4 and f_y(1, 3) = -2. Explain what these two numbers mean if you move a small distance from (1, 3) in the positive x direction or the positive y direction.