Introduction to Partial Derivatives
A partial derivative is a derivative of a function with multiple variables, concerning just one of those variables. For a function f(x,y), the partial derivative with respect to x is written as ∂f/∂x, and the partial derivative with respect to y is ∂f/∂y.
Partial derivatives differ from regular derivatives in that they allow you to find the rate of change of a multivariable function with respect to one variable while treating the other variables as constants. This is necessary for functions of multiple variables because the rate of change can be different along the x, y, and z axes.
For example, consider the function f(x,y) = x^2 + xy + y^2. The partial derivative with respect to x is ∂f/∂x = 2x + y. This gives the rate of change in the x direction. The partial derivative with respect to y is ∂f/∂y = x + 2y, which gives the rate of change in the y direction. These rates of change can be different at a given point (x,y).
Partial derivatives allow us to analyze functions of multiple variables by looking at each dimension independently. This is essential for optimization, analyzing rates of change, approximations, and many other applications.
Partial Derivative Notation
The notation used for partial derivatives is similar to regular derivatives but uses a different symbol. For a function f(x,y), the partial derivative with respect to x is written:
∂f/∂x
And the partial derivative with respect to y is written:
∂f/∂y
The symbol ∂ (called “partial”) replaces the d symbol used for regular derivatives. This helps distinguish that it’s a partial derivative taken with respect to only one variable, rather than the total derivative.
For example, if f(x,y) = x^2 + 3xy + y^2, then:
∂f/∂x = 2x + 3y
And
∂f/∂y = 3x + 2y
The partial derivative notation clearly shows which variable we are taking the derivative with respect to. Proper use of the notation is important when working with multivariable functions.
Taking Simple Partial Derivatives
Taking partial derivatives of basic functions with respect to one variable is straightforward. We simply treat the other variable as a constant and take the regular derivative with respect to the desired variable.
For example, consider the function f(x,y) = x2 + 3xy + y. To take the partial derivative with respect to x, we treat y as a constant and take the derivative of each term:
∂f/∂x = 2x + 3y
To take the partial derivative with respect to y, we treat x as a constant and take the derivative of each term:
∂f/∂y = 3x + 1
As another example, consider the function g(x,y) = sin(x) + ey. The partial derivatives are:
∂g/∂x = cos(x)
∂g/∂y = ey
These examples demonstrate the basic process of taking partial derivatives of functions with respect to one variable. We simply derive as normal while treating other variables as constants. With practice, this process becomes second nature.
Let’s now look at some more complex examples involving products and composite functions.
Common Rules of Partial Differentiation
Taking partial derivatives often involves applying common rules like the product rule, chain rule, and quotient rule. These rules help simplify the process of finding partial derivatives.
The product rule states that if f(x,y) = u(x,y) * v(x,y), then:
∂f/∂x = ∂u/∂x * v + u * ∂v/∂x
For example, if f(x,y) = x^2 * e^y, then applying the product rule gives:
∂f/∂x = (2x) * e^y + x^2 * (e^y)
The chain rule can be applied when the function is a composite function. For example, if f(x,y) = sin(x^2 + y), then:
∂f/∂x = cos(x^2 + y) * 2x
The quotient rule is similar to single-variable calculus:
∂(u/v)/∂x = (v * ∂u/∂x – u * ∂v/∂x) / v^2
Applying the quotient rule, if f(x,y) = (x^2 + y) / (x + y^2), then:
∂f/∂x = (x + y^2) * (2x) – (x^2 + y) * (1) / (x + y^2)^2
These basic rules can be combined and applied multiple times when taking partial derivatives. With practice, applying these rules becomes second nature when differentiating functions with multiple variables.
Higher Order Partial Derivatives
Taking higher-order partial derivatives means taking the derivative more than once with respect to the same variable. For example, taking the second partial derivative of a function means taking the derivative twice.
The notation for higher-order partial derivatives indicates how many times the derivative is taken with respect to each variable. For the second partial derivative with respect to x, we write:
∂2f/∂x2
For the third partial derivative with respect to x, we write:
∂3f/∂x3
And so on for higher orders. We can also take mixed higher-order derivatives, like the second-order mixed derivative:
∂2f/∂x∂y
This means take the derivative with respect to x first, then take the derivative with respect to y.
For example, consider the function f(x,y) = x2 + 3xy + y2. To find the second order partial derivative with respect to x, we take:
∂2f/∂x2 = ∂/∂x (2x + 3y) = 2
The third order derivative with respect to y is:
∂3f/∂y3 = ∂/∂y (2) = 0
And the mixed second order derivative is:
∂2f/∂x∂y = ∂/∂y (2x + 3y) = 3
So in summary, taking higher order derivatives just means applying the derivative operator multiple times with respect to the same variable(s). The notation clearly indicates the order and variables.
[1] https://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspxGeometric Interpretation of Partial Derivatives
Partial derivatives give the rates of change of a function along the input variables. We can visualize this geometrically on 3D plots of multivariable functions.
For example, consider the function f(x,y) = x^2 + y^2. The partial derivative ∂f/∂x represents the rate of change of f as we move along the x-axis, keeping y fixed. Geometrically, this is the slope of the tangent line to the function along the x-axis slice.
Similarly, ∂f/∂y gives the rate of change along the y-axis, which is the slope of the tangent line along the y-axis slice. We can visualize the partial derivatives at a given point (x,y) on a 3D plot of the function.
In general, the partial derivative ∂f/∂x represents the slope of the tangent plane to the function f(x,y) in the x direction. ∂f/∂y gives the slope in the y direction. Understanding partial derivatives geometrically in this way helps provide intuition about their meaning.
For more examples visualizing partial derivatives on 3D plots, see: Geometry of partial derivatives
Physical Examples and Applications
Partial derivatives have many important applications in physics, economics, and other fields. In physics, partial derivatives are used to describe rates of change in systems with multiple variables.
For example, in fluid mechanics the velocity field of a fluid flow can be described using the partial derivatives of the velocity components. The continuity equation relates the partial derivatives of the velocity components to describe how mass is conserved in the flow. Partial derivatives of temperature can describe heat flux and temperature gradients in thermodynamic systems.
In economics, partial derivatives describe how a function changes in response to variations in its arguments. For example, the partial derivative of a cost function with respect to quantity produced gives the marginal cost – how cost changes with an increase in quantity. The partial derivative of a revenue function with respect to price gives the marginal revenue.
Optimization problems in economics often involve taking partial derivatives of profit or utility functions and setting them equal to zero to find optimal values. Implicit differentiation using partial derivatives is also commonly used in economics.
Overall, partial derivatives allow rates of change to be analyzed in systems with multiple independent variables. They are invaluable mathematical tools with applications across science and engineering.
Using Partial Derivatives for Optimization
Partial derivatives can be used to find the optimal points of a function with multiple variables. This is very useful in economics, physics, and other fields where we want to optimize some quantity.
To find the optimal points, we take the partial derivative of the function with respect to each variable, set them equal to zero, and solve the resulting system of equations. The points that satisfy the equations are candidates for optima.
For example, consider the function f(x,y) = x^2 + y^2. Taking the partial derivatives:
∂f/∂x = 2x
∂f/∂y = 2y
Setting these equal to zero gives x = 0 and y = 0. So the point (0,0) is a critical point and a candidate optimum.
We can also optimize functions with constraints using the method of Lagrange multipliers. This involves introducing a new variable to account for the constraint equation. The partial derivatives of the Lagrangian function are set equal to zero and solved.
For example, to optimize f(x,y) = x + y subject to x^2 + y^2 = 1:
Construct the Lagrangian: L(x,y,λ) = f(x,y) – λ(g(x,y))
Where g(x,y) = x^2 + y^2 – 1 = 0 is the constraint equation.
Taking partial derivatives and setting equal to zero:
∂L/∂x = 1 – 2λx = 0
∂L/∂y = 1 – 2λy = 0
∂L/∂λ = x^2 + y^2 – 1 = 0
Solving this system gives the optimal point at (1/√2, 1/√2).
So Lagrange multipliers allow optimizing constrained multivariable functions, very useful in economics and physics.
For more examples, see: https://economics.uwo.ca/math/resources/calculus-multivariable-functions/5-partial-derivatives-optimization-constraints/content/
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of an implicitly defined function. An implicitly defined function is one where the relationship between the variables is defined by an equation, rather than having one variable explicitly written as a function of the others.
For example, consider the equation x^2 + y^2 = 25. This implicitly defines y as a function of x, but we don’t have y explicitly written as f(x). To find dy/dx, we can use implicit differentiation.
The key steps are:
- Take the partial derivative of both sides of the equation with respect to the variable of interest. In this case, take the partial of both sides with respect to x.
- Solve the resulting equation for the derivative you want. Here, we would solve for dy/dx.
Applying this procedure to our example equation:
Take the partial of both sides w.r.t. x: 2x + 2y dy/dx = 0
Solve for dy/dx: dy/dx = -x/y
So the implicit differentiation yielded the derivative dy/dx = -x/y for this implicitly defined function.
The chain rule can also be used when implicitly differentiating functions of multiple variables. For example, consider the equation x^2 + y^2 = z^2. Taking the partial of both sides with respect to x using the chain rule gives:
2x + 2y dy/dx = 2z dz/dx
Then dy/dx and dz/dx can be solved for in terms of x, y, and z.[1]
Summary
In this article, we covered the key concepts and applications of partial derivatives. To recap:
– A partial derivative measures how a function changes as you vary one variable while holding others constant. It allows us to analyze functions of multiple variables.
– The notation for partial derivatives is ∂f/∂x, ∂f/∂y, etc. The partial symbol reminds us we are taking a “partial” derivative.
– To find partial derivatives, treat other variables like constants and take the regular derivative. We applied rules like the product rule and chain rule.
– Higher order partials involve taking partials repeatedly. Notation uses multiple ∂ symbols.
– Geometrically, partials show instantaneous rates of change and tangent lines along the coordinate axes.
– We saw applications in optimization, economics, physics, etc. Partials describe sensitivity and localized rates of change.
– Techniques like implicit differentiation also rely on partial derivatives.
In summary, partial derivatives are essential for analyzing functions of multiple variables. They have wide-ranging applications in math, science, engineering, and economics. This article covered the key concepts and skills needed to evaluate partial derivatives confidently.
