Partial Derivative

Introduction  

Partial derivatives are used in mathematics when dealing with functions that involve more than one variable. They help us understand how a function changes with respect to one variable while keeping the other variables constant. In simple terms, a partial derivative measures the rate of change of a multivariable function with respect to a single variable. This concept plays a crucial role in fields such as physics, engineering, economics, and computer science.

 

Table of Contents

• What is a Partial Derivative?
• Partial Derivative Definition
• Symbol of Partial Derivative
• Partial Derivative Formula
• Understanding Partial Differentiation
• Rules of Partial Derivatives
  • Power Rule
  • Product Rule
  • Quotient Rule
  • Chain Rule
• Partial Derivative of Natural Log
• Partial Derivative Examples
• Conclusion
• FAQs On Partial Derivative

 

What is a Partial Derivative?  

The partial derivative of a function is calculated with respect to one variable while treating all other variables as constants. When a function depends on several variables, a partial derivative helps us see how the function behaves when only one variable changes.  

For example, if we have a function f(x, y), the partial derivative of f with respect to x means we differentiate f with respect to x, treating y as constant.  

It measures how a function changes when only one of the input variables changes. This is important for dealing with multivariable functions.  

 

Partial Derivative Definition  

 The partial derivative definition for a function f(x, y) with respect to x is given by the following limit: 

∂f/∂x = lim(Δx → 0) [f(x + Δx, y) − f(x, y)] / Δx  

Similarly, the partial derivative with respect to y is:  

∂f/∂y = lim(Δy → 0) [f(x, y + Δy) − f(x, y)] / Δy  

This mathematical definition indicates that we slightly change one variable while keeping the others fixed and see how the function value changes.  

 

Symbol of Partial Derivative  

The symbol for the partial derivative is the rounded letter '∂', not the regular 'd' used in ordinary derivatives. Some common notations for the partial derivative of a function f with respect to x are:  

∂f/∂x  

fx  

∂/∂x (f)  

All of these mean the same thing: taking the derivative of f with respect to x while keeping the other variables constant.  

 

Partial Derivative Formula  

The partial derivative formula is used to differentiate multivariable functions. To find the partial derivative of a function f(x, y, z), we treat all other variables as constants except for the one we are differentiating with respect to.  

Suppose we have f(x, y) = x²y + y³  

Then,  

• ∂f/∂x = 2xy (treat y as constant)  

• ∂f/∂y = x² + 3y² (treat x as constant). 

This is the basic partial derivative formula, and it can be applied to various functions based on their complexity.  

 

Understanding Partial Differentiation  

When we study functions that depend on more than one variable, such as f(x, y) or f(x, y, z), we often want to know how the function changes if only one variable is varied while keeping the others fixed. This process is called partial differentiation.

A partial derivative is simply the derivative of a multivariable function with respect to one of its variables, while the other variables are treated as constants.

For example:
f(x, y) = x²y + 3y

If we differentiate with respect to x, treating y as a constant, we get:
∂f/∂x = 2xy

If we differentiate with respect to y, treating x as a constant, we get:
∂f/∂y = x² + 3

This method is very useful in real-life problems involving physics, engineering, and economics, where many variables affect the outcome but we may want to study the effect of just one at a time.

 

Rules of Partial Derivatives

When calculating partial derivatives, there are several important rules to remember. These include the Power Rule, Product Rule, Quotient Rule, and Chain Rule. Each of these rules helps simplify complex functions The only difference is that we treat all other variables as constants while differentiating with respect to one variable.

Power Rule

If f(x, y) = x^n * y^m, then  

∂f/∂x = n * x^(n-1) * y^m  

∂f/∂y = m * x^n * y^(m-1)  

This rule applies when variables are raised to powers.

 

Product Rule

If f(x, y) = u(x, y) * v(x, y), then the partial derivative of f with respect to x is:  

∂f/∂x = ∂u/∂x * v + u * ∂v/∂x  

We handle it just like the product rule in single-variable calculus, but each term is a partial derivative.

 

Quotient Rule

If f(x, y) = u(x, y) / v(x, y), then:  

∂f/∂x = [v * ∂u/∂x - u * ∂v/∂x] / v²  

This rule is useful for ratios of functions.

 

Chain Rule

If z = f(u), and u is a function of x and y, then:  

∂z/∂x = df/du * ∂u/∂x  

∂z/∂y = df/du * ∂u/∂y  

We use the chain rule when one function depends on another function that in turn depends on other variables.

 

Partial Derivative of Natural Log

To find the partial derivative of the natural log, we use the identity that d/dx (ln x) = 1/x. For multivariable functions, the same idea applies.

Example:  

Let f(x, y) = ln(xy)  

Then,  

∂f/∂x = 1/(xy) * y = 1/x  

∂f/∂y = 1/(xy) * x = 1/y  

So, the partial derivative of natural log functions follows the same rule as in single-variable calculus, but we must use the chain rule when the argument is a product or composition.

 

Partial Derivative Examples

Example 1:
f(x, y) = x²y + y³
∂f/∂x = 2xy
∂f/∂y = x² + 3y²

 

Example 2:
f(x, y, z) = x²yz + ln(x + y)
∂f/∂x = 2xyz + 1/(x + y)
∂f/∂y = x²z + 1/(x + y)
∂f/∂z = x²y

 

Example 3:
f(x, y) = (x² + y²) / (x − y)
Using the quotient rule:
∂f/∂x = [(2x)(x − y) − (x² + y²)(1)] / (x − y)²

 

Example 4:
f(x, y) = ln(x² + y²)
∂f/∂x = (1 / (x² + y²)) * 2x = 2x / (x² + y²)
∂f/∂y = (1 / (x² + y²)) * 2y = 2y / (x² + y²)

These partial derivative examples help understand how the concept is applied in real problems and why rules like the chain rule and quotient rule are necessary.

 

Conclusion  

The partial derivative is an important mathematical concept used to explore how multivariable functions behave. It shows how a function changes in relation to one variable while keeping the others fixed. Knowing what a partial derivative is, along with its definition, formula, and rules like the product rule, quotient rule, and chain rule, is crucial for solving more complex math and applied problems. The partial derivative of natural log and partial derivative examples often appear on exams and help connect theory with practical use.  

If you are dealing with functions that have more than one variable, learning the partial derivative is an essential step in calculus and mathematical modeling.

 

Frequently Asked Questions on Partial Derivative

1. What is an example of a partial derivative?

Answer: An example of a partial derivative:
If f(x, y) = x²y + y³, then the partial derivative of f with respect to x is ∂f/∂x = 2xy.
This means we treat y as constant while differentiating with respect to x.

 

2. What is the letter ∂ called?

Answer: The letter ∂ is called "partial derivative symbol" or "del". It is used specifically for partial derivatives in multivariable calculus.

 

3. What is the difference between ∂ and d?

Answer: The symbol ∂ is used for partial derivatives (when functions have more than one variable), while d is used for ordinary derivatives (functions with only one variable).

 

4. What is the formula for the partial derivative?

Answer: The formula for a partial derivative of a function f(x, y) with respect to x is:
∂f/∂x = limit as h→0 of [f(x + h, y) - f(x, y)] / h

 

5. What is the formula for derivatives?

Answer: The basic formula for the derivative of a function f(x) is:
f′(x) = limit as h→0 of [f(x + h) - f(x)] / h

 

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