Determine Which Function Is The Derivative Of The Other

Determining Which Function is the Derivative of the Other: A Comprehensive Guide

Determining whether one function is the derivative of another is a fundamental concept in calculus. This ability is crucial for understanding rates of change, optimization problems, and the behavior of functions. This comprehensive guide will explore various techniques and strategies to solve this problem, progressing from simple examples to more complex scenarios. We'll cover both analytical and graphical approaches, providing a solid foundation for tackling derivative problems.

Understanding Derivatives: A Quick Recap

Before we delve into the process of identifying derivative relationships, let's briefly review the core concept of a derivative. The derivative of a function, denoted as f'(x) or df/dx, represents the instantaneous rate of change of the function at a specific point. Geometrically, it represents the slope of the tangent line to the function's graph at that point.

The derivative is found through a process called differentiation, which involves applying various rules and techniques depending on the function's complexity. Common differentiation rules include the power rule, product rule, quotient rule, and chain rule.

Method 1: Direct Differentiation

The most straightforward method to determine if one function is the derivative of another is to directly differentiate the suspected "original" function and compare the result to the suspected "derivative" function.

Example 1: Simple Polynomial

Let's consider two functions: f(x) = x² + 2x and g(x) = 2x + 2. We suspect g(x) is the derivative of f(x). Let's differentiate f(x):

f'(x) = d/dx (x² + 2x) = 2x + 2

Since f'(x) = g(x), we conclude that g(x) is indeed the derivative of f(x).

Example 2: Involving Trigonometric Functions

Consider f(x) = sin(x) and g(x) = cos(x). Is g(x) the derivative of f(x)?

Differentiating f(x):

f'(x) = d/dx (sin(x)) = cos(x)

Therefore, g(x) is the derivative of f(x).

Example 3: Applying the Chain Rule

Let f(x) = (x² + 1)³ and g(x) = 6x(x² + 1)². Is g(x) the derivative of f(x)?

Applying the chain rule to differentiate f(x):

f'(x) = d/dx [(x² + 1)³] = 3(x² + 1)² * d/dx(x² + 1) = 3(x² + 1)² * 2x = 6x(x² + 1)²

Since f'(x) = g(x), g(x) is the derivative of f(x).

Method 2: Antidifferentiation (Integration)

If we suspect that function g(x) is the derivative of function f(x), we can find the antiderivative (indefinite integral) of g(x) and check if the result is f(x), possibly differing by a constant.

Example 4: Using Antidifferentiation

Let's assume g(x) = 3x² + 4x and we want to determine if there exists a function f(x) such that f'(x) = g(x). We find the antiderivative of g(x):

∫(3x² + 4x) dx = x³ + 2x² + C (where C is the constant of integration)

Therefore, f(x) = x³ + 2x² + C. Any function of this form (varying only in the value of C) would have g(x) as its derivative.

Method 3: Graphical Analysis

Graphical analysis can offer a visual approach to this problem, although it's less precise than analytical methods.

Interpreting the Graph:

• Slope of the Tangent: The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point. If you can visually estimate the slopes of the tangent lines to the graph of one function and compare them to the y-values of the second function, you might infer a derivative relationship.

• Increasing/Decreasing Behavior: If one function is increasing (positive derivative) where the other is positive, and decreasing (negative derivative) where the other is negative, it's suggestive of a derivative relationship. However, this is not conclusive proof.

Limitations of Graphical Analysis:

Graphical analysis is subjective and limited by the accuracy of the graph and the ability to visually estimate slopes. It's best used as a preliminary check or to support analytical findings.

Handling More Complex Functions

For functions involving more intricate combinations of elementary functions, we need to utilize a combination of differentiation rules and techniques.

Example 5: Functions with Exponential and Logarithmic Components

Let's consider f(x) = ln(x) and g(x) = 1/x. Is g(x) the derivative of f(x)?

Differentiating f(x):

f'(x) = d/dx [ln(x)] = 1/x

Hence, g(x) is the derivative of f(x).

Example 6: Involving Implicit Differentiation

When dealing with implicitly defined functions, the process becomes slightly more involved. Implicit differentiation is required to find the derivative.

Consider the equation x² + y² = 25. We can express y as a function of x (y = ±√(25 - x²)) and then differentiate explicitly, or we can use implicit differentiation. Differentiating both sides with respect to x:

2x + 2y(dy/dx) = 0

Solving for dy/dx:

dy/dx = -x/y

This shows the derivative of y with respect to x as a function of both x and y. This illustrates how implicit differentiation handles relationships where y isn't explicitly defined as a function of x.

Dealing with Constants of Integration

Remember that the antiderivative of a function is not unique; it's defined up to a constant of integration (C). This means if G(x) is an antiderivative of g(x), then G(x) + C is also an antiderivative for any constant C. This must be considered when comparing functions.

Advanced Techniques and Considerations

For highly complex functions, numerical methods, symbolic computation software (like Mathematica or Maple), or advanced calculus techniques might be necessary to definitively determine the derivative relationship.

Conclusion

Determining whether one function is the derivative of another is a crucial skill in calculus. By mastering the techniques discussed—direct differentiation, antidifferentiation, and graphical analysis—you can effectively tackle various problems involving derivative relationships. Remember that while graphical analysis can provide insights, analytical methods offer definitive answers. Always consider the constant of integration when working with antiderivatives. With practice and a solid understanding of differentiation and integration rules, you'll become proficient in identifying these fundamental relationships between functions.