Solutions to Examples on Partial Derivatives
Introduction to Partial Derivatives
Partial derivatives are fundamental concepts in multivariable calculus that measure how a function changes as one variable changes, while all other variables remain constant. Understanding partial derivatives is essential for many fields including physics, engineering, economics, and other disciplines that deal with functions of multiple variables.
Basic Concepts
For a function f(x, y), the partial derivative with respect to x (denoted as f/x or f) measures the rate of change of f in the x-direction while keeping y constant. Similarly, the partial derivative with respect to y (denoted as f/y or f) measures the rate of change of f in the y-direction while keeping x constant.
The fundamental rules for finding partial derivatives include:
- To find f/x, treat all variables except x as constants and differentiate with respect to x.
- To find f/y, treat all variables except y as constants and differentiate with respect to y.
- All standard differentiation rules (power rule, product rule, quotient rule, chain rule) apply to partial derivatives.
Example 1: Finding Partial Derivatives of a Basic Function
Problem: Find the partial derivatives f/x and f/y for the function f(x, y) = 3xy + 5xy - 2y + 7x.
Solution:
Step 1: To find f/x, we treat y as a constant and differentiate with respect to x:
Applying the power rule to each term:
Step 2: To find f/y, we treat x as a constant and differentiate with respect to y:
Again applying the power rule to each term:
Answer: f/x = 6xy + 5y + 7, f/y = 3x + 15xy - 2
Example 2: Partial Derivatives of an Exponential Function
Problem: Find the partial derivatives f/x and f/y for the function f(x, y) = e^(xy) - ln(x + y).
Solution:
Step 1: To find f/x, we treat y as a constant and differentiate with respect to x:
For the first term, we use the chain rule:
For the second term:
Combining these results:
Step 2: To find f/y, we treat x as a constant and differentiate with respect to y:
For the first term, we use the chain rule:
For the second term:
Combining these results:
Answer: f/x = 2xy * e^(xy) - 1/(x + y), f/y = x * e^(xy) - 1/(x + y)
Example 3: Partial Derivatives of a Function with Three Variables
Problem: Find f/x, f/y, and f/z for the function f(x, y, z) = xyz + xyz + xyz.
Solution:
Step 1: To find f/x, we treat y and z as constants and differentiate with respect to x:
Applying the power rule to each term:
Step 2: To find f/y, we treat x and z as constants and differentiate with respect to y:
Applying the power rule to each term:
Step 3: To find f/z, we treat x and y as constants and differentiate with respect to z:
Applying the power rule to each term:
Answer: f/x = yz(2x + y + z), f/y = xz(x + 2y + z), f/z = xy(x + y + 2z)
Example 4: Second-Order Partial Derivatives
Problem: For the function f(x, y) = xy - 2xy + 3xy, find all second-order partial derivatives.
Solution:
Step 1: First, we find the first-order partial derivatives:
Step 2: Now, we find the second-order partial derivatives:
Note: As expected, f/xy = f/yx, which is a consequence of Clairaut's theorem.
Answer: f/x = 6xy - 4y, f/y = 2x - 24xy, f/xy = f/yx = 6xy - 16xy + 3
Example 5: Using the Chain Rule with Partial Derivatives
Problem: Let f(x, y) = x + y. If x = uv and y = u/v, find f/u and f/v using the chain rule.
Solution:
Step 1: First, find the partial derivatives of f with respect to x and y:
Step 2: Find the partial derivatives of x and y with respect to u and v:
Step 3: Apply the chain rule for f/u:
Step 4: Substitute x = uv and y = u/v:
Step 5: Apply the chain rule for f/v:
Step 6: Substitute x = uv and y = u/v:
Answer: f/u = 4uv + 3u/(v), f/v = 2uv - 3u/(v)
Example 6: Directional Derivatives
Problem: Find the directional derivative of the function f(x, y) = xy - 3xy at the point (1, 2) in the direction of the vector v = 3i + 4j.
Solution:
Step 1: First, calculate the gradient of f at point (1, 2):
Evaluating at (1, 2):
Step 2: Find the unit vector in the direction of v = 3i + 4j:
Step 3: Calculate the directional derivative:
Answer: The directional derivative is 66/5.
Applications of Partial Derivatives
Partial derivatives have numerous applications across various scientific and engineering disciplines:
- Thermodynamics: Used to describe how state variables like pressure, volume, temperature, and entropy change in thermodynamic processes.
- Fluid Mechanics: Essential for analyzing velocity fields, pressure distributions, and flow patterns in fluids.
- Economics: Critical for calculating marginal utilities, marginal costs, and analyzing multi-factor market dynamics.
- Optimization: Used in finding maxima and minima of functions with multiple variables through techniques like Lagrange multipliers.
- Differential Equations: Form the basis for partial differential equations that describe many physical phenomena.
Conclusion
Partial derivatives are powerful mathematical tools that extend the concept of derivatives to functions of multiple variables. Through the examples and solutions discussed above, we see how the basic rules of differentiation apply while considering other variables as constants. Mastering these techniques is essential for advanced work in mathematics, physics, engineering, economics, and many other fields that require analyzing multivariate systems. Regular practice with diverse problems helps develop the skill and intuition needed to effectively apply partial derivatives in various contexts.
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