Partial Derivatives Examples And A Quick Review of Implicit Differentiation
Partial derivatives are a fundamental concept in multivariable calculus. They allow us to understand how a function changes with respect to one variable while keeping others constant. This page provides examples of partial derivatives and includes a brief review of implicit differentiation as a prerequisite concept.
Review of Implicit Differentiation
Before diving into partial derivatives, let's briefly review implicit differentiation. In single-variable calculus, we often deal with functions defined explicitly, like y = f(x). However, sometimes equations are given in an implicit form where y is not isolated on one side.
To find dy/dx for such equations, we use implicit differentiation:
1. Differentiate both sides of the equation with respect to x.
2. Whenever we differentiate a term containing y, we apply the chain rule and multiply by dy/dx.
3. Solve for dy/dx.
Example 1: Implicit Differentiation
For the equation x2 + y2 = 25:
Applying implicit differentiation:
2x + 2y(dy/dx) = 0
Solving for dy/dx:
2y(dy/dx) = -2x
dy/dx = -2x/2y = -x/y
Introduction to Partial Derivatives
When working with functions of multiple variables, we want to know how the function changes as we vary one variable while keeping others constant. This is where partial derivatives come into play.
f/x (read as "partial f with respect to x") measures how f changes as x changes, with y held constant.
f/y (read as "partial f with respect to y") measures how f changes as y changes, with x held constant.
Calculating Partial Derivatives
To find partial derivatives, we treat all variables except the one we're differentiating with respect to as constants. We then apply the usual differentiation rules.
Example 2: Basic Partial Derivatives
For the function f(x,y) = 3x2 + 2xy + y3:
To find f/x:
Treat y as a constant and differentiate with respect to x:
f/x = 6x + 2y
To find f/y:
Treat x as a constant and differentiate with respect to y:
f/y = 2x + 3y2
Higher Order Partial Derivatives
Just as we can take higher order derivatives of single-variable functions, we can take higher order partial derivatives:
2f/y2 = /y (f/y)
2f/xy = /x (f/y)
2f/yx = /y (f/x)
Example 3: Second Order Partial Derivatives
For the function f(x,y) = x3y2 + 2x + y4:
First order partial derivatives:
f/x = 3x2y2 + 2
f/y = 2x3y + 4y3
Second order partial derivatives:
2f/x2 = 6xy2
2f/y2 = 2x3 + 12y2
2f/xy = 6x2y
2f/yx = 6x2y
Chain Rule for Partial Derivatives
The chain rule extends to multivariable functions. If z = f(x,y) and x and y are functions of t (i.e., x = x(t) and y = y(t)), then:
Example 4: Chain Rule for Partial Derivatives
For the function z = x2y + 3xy2, where x = t2 + 1 and y = t3:
First, find the partial derivatives:
z/x = 2xy + 3y2
z/y = x2 + 6xy
Next, find the derivatives of x and y with respect to t:
dx/dt = 2t
dy/dt = 3t2
Using the chain rule:
dz/dt = (2xy + 3y2)(2t) + (x2 + 6xy)(3t2)
Substituting x = t2 + 1 and y = t3:
dz/dt = (2(t2+1)(t3) + 3(t3)2)(2t) + ((t2+1)2 + 6(t2+1)(t3))(3t2)
Gradient and Directional Derivative
The gradient of a function f(x,y) is a vector of its partial derivatives:
The directional derivative of f at a point (a,b) in the direction of a unit vector u = (u1, u2) is:
Example 5: Gradient and Directional Derivative
For the function f(x,y) = x2 + y2:
The gradient is:
f = (2x, 2y)
At the point (1,2):
f(1,2) = (2, 4)
The directional derivative at (1,2) in the direction of u = (1/2, 1/2) is:
Duf(1,2) = f(1,2) u = (2, 4) (1/2, 1/2) = 6/2 = 32
Tangent Planes and Linear Approximations
The tangent plane to the surface z = f(x,y) at the point (a,b,f(a,b)) is:
This tangent plane provides a linear approximation to the function near the point (a,b).
Example 6: Tangent Plane
For the function f(x,y) = x2 + 3xy - y2 at the point (1,2):
First, evaluate the function at (1,2):
f(1,2) = 12 + 3(1)(2) - 22 = 1 + 6 - 4 = 3
Find the partial derivatives:
f/x = 2x + 3y
f/y = 3x - 2y
Evaluate the partial derivatives at (1,2):
f/x|(1,2) = 2(1) + 3(2) = 8
f/y|(1,2) = 3(1) - 2(2) = -1
The equation of the tangent plane is:
z = 3 + 8(x-1) - 1(y-2)
Simplifying:
z = 8x - y - 3
Applications of Partial Derivatives
Partial derivatives have numerous applications in science, engineering, and economics:
- Physics: In thermodynamics, partial derivatives describe how physical quantities like temperature, pressure, and volume relate to each other.
- Engineering: Engineers use partial derivatives to analyze stress and strain in materials, optimizing designs.
- Economics: Marginal analysis in economics uses partial derivatives to understand how changes in one variable affect outcomes when other variables are held constant.
- Machine Learning: Training algorithms often involves the calculation of partial derivatives in optimization processes like gradient descent.
Example 7: Economics Application
Consider a Cobb-Douglas production function: P(L,K) = ALK
Where:
- P is the production output
- L is labor input
- K is capital input
- A, , and are constants with , > 0
The partial derivative with respect to labor (P/L) gives the marginal product of labor, showing how much production changes when labor increases by one unit while capital remains constant.
P/L = AL(-1)K
Similarly, the partial derivative with respect to capital (P/K) gives the marginal product of capital:
P/K = ALK(-1)
Conclusion
Partial derivatives extend the concept of derivatives to functions of multiple variables. They allow us to analyze how a function changes with respect to one variable while keeping others constant. Together with implicit differentiation, partial derivatives form a powerful toolkit for solving problems in calculus and its applications across various disciplines.
Understanding these concepts is crucial for tackling more advanced topics in multivariable calculus, including optimization of functions of several variables, multiple integrals, and vector calculus.
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