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.
Understanding Calculus: Partial Derivatives
The rule for calculating a partial derivative is simple, but careful attention is needed. Imagine a function that gives height as three x squared y plus five y. To find the partial derivative with respect to x, freeze y at whatever value it has.
The term three x squared y behaves like a constant multiplier times x squared, so its derivative is six x y. The term five y has no changing x in it, so its derivative is zero. The result is six x y.
To differentiate with respect to y instead, freeze x. The result becomes three x squared plus five. Each answer is usually still a function of both inputs because the frozen variable can have different values at different locations.
Partial derivatives have units, and units help students check whether an answer makes sense. If a hill map gives elevation in metres while horizontal position is measured in kilometres, the partial derivative with respect to eastward position has units of metres per kilometre. A positive value means elevation rises when moving east at that location.
A negative value means it falls. Its size tells how rapidly the elevation changes for a small movement.
The derivative is a local measurement. It describes behaviour close to one point, not necessarily across an entire hill or map.
In science, partial derivatives separate effects that would otherwise be mixed together. Pressure in a gas can depend on volume and temperature. The partial derivative with respect to volume describes the pressure change from a tiny volume change when temperature is controlled.
The partial derivative with respect to temperature describes a different experiment where volume is controlled. This distinction matters because real experiments often require some quantities to remain fixed.
In economics, a cost might depend on the number of products made and the price of raw materials. A partial derivative can show the effect of producing one more item before the raw material price changes.
Partial derivatives become especially useful when finding high or low points of a surface. At an interior maximum, minimum, or flat saddle point, both first partial derivatives are often zero. That condition only identifies candidates, so more work is needed to classify them.
Second partial derivatives describe how the surface curves near the candidate point. Students should practise taking mixed partial derivatives too. For many smooth functions, differentiating first with respect to x then y gives the same result as differentiating first with respect to y then x.
Errors commonly come from forgetting which variable is frozen, differentiating a constant term, or evaluating at a point too early. Keep the variables until the derivative is complete, then substitute the point values.
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.