What Is The Derivative Of 1 1 X

What is the Derivative of 1/(1+x)? A Comprehensive Guide

The question "What is the derivative of 1/(1+x)?" seems simple, but understanding its solution unlocks a deeper understanding of calculus, particularly the power rule, the chain rule, and even the concept of infinite series. This article will explore various methods of finding the derivative, delve into its applications, and touch upon related concepts.

Understanding the Basics: Derivatives and the Power Rule

Before tackling the specific problem, let's refresh our understanding of derivatives. The derivative of a function, f(x), denoted as f'(x) or df/dx, represents the instantaneous rate of change of the function with respect to x. Geometrically, it represents the slope of the tangent line to the curve of the function at a given point.

One of the fundamental rules of differentiation is the power rule:

The derivative of xⁿ is nx^n-1, where n is a constant.

However, our function, 1/(1+x), isn't directly in the form xⁿ. We need to manipulate it before applying the power rule.

Method 1: Rewriting the Function using Negative Exponents

We can rewrite 1/(1+x) using negative exponents:

1/(1+x) = (1+x)^-1

Now, we can apply the chain rule and the power rule. The chain rule states that the derivative of a composite function is the derivative of the outer function (with the inside function left alone) times the derivative of the inner function.

Let's break it down:

• Outer function: u^-1, where u = (1+x)
• Inner function: u = (1+x)

Applying the chain rule:

d/dx[(1+x)^-1] = -1(1+x)^-2 * d/dx(1+x)

The derivative of (1+x) with respect to x is simply 1. Therefore:

d/dx[(1+x)^-1] = -1(1+x)^-2 * 1 = -1/(1+x)²

This is our final answer using this method. It's crucial to remember the chain rule when dealing with composite functions.

Method 2: Using the Quotient Rule

Another approach involves using the quotient rule, which is specifically designed for differentiating functions in the form f(x)/g(x). The quotient rule states:

d/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]²

In our case:

• f(x) = 1
• g(x) = 1 + x

Applying the quotient rule:

d/dx[1/(1+x)] = [(1+x)(d/dx(1)) - (1)(d/dx(1+x))] / (1+x)²

The derivative of 1 is 0, and the derivative of (1+x) is 1. Substituting these values:

d/dx[1/(1+x)] = [(1+x)(0) - (1)(1)] / (1+x)² = -1 / (1+x)²

Again, we arrive at the same result: -1/(1+x)²

Method 3: Implicit Differentiation (An Advanced Approach)

While less direct for this specific problem, implicit differentiation offers a valuable perspective. Let's assume y = 1/(1+x). We can rewrite this as:

y(1+x) = 1

Now, differentiate both sides with respect to x:

d/dx[y(1+x)] = d/dx(1)

Using the product rule on the left-hand side (which is another essential differentiation technique):

y(d/dx(1+x)) + (1+x)(dy/dx) = 0

Simplifying:

y(1) + (1+x)(dy/dx) = 0

Solving for dy/dx:

dy/dx = -y / (1+x)

Substitute y = 1/(1+x):

dy/dx = [-1/(1+x)] / (1+x) = -1/(1+x)²

This demonstrates the flexibility of implicit differentiation, particularly useful when dealing with more complex equations.

Applications of the Derivative: Understanding the Rate of Change

The derivative -1/(1+x)² provides crucial information about the original function 1/(1+x). Specifically, it tells us the instantaneous rate of change at any point x.

• Slope of the Tangent: At any given x-value, the derivative gives the slope of the tangent line to the curve y = 1/(1+x). This is useful in various applications, including optimization problems and curve sketching.
• Increasing/Decreasing Intervals: By analyzing the sign of the derivative, we can determine where the function is increasing or decreasing. Since -1/(1+x)² is always negative (except when the denominator is zero, resulting in a vertical asymptote at x=-1), the function 1/(1+x) is always decreasing where it is defined.
• Concavity: The second derivative provides information about the concavity of the function (whether it's curving upwards or downwards). Finding the second derivative (which involves differentiating -1/(1+x)² again) allows for a more detailed analysis of the function's behavior.

Connection to Infinite Series (Taylor and Maclaurin Series)

The function 1/(1+x) is closely related to the geometric series. The geometric series formula is:

1 + x + x² + x³ + ... = 1/(1-x) for |x| < 1

By substituting -x for x, we get:

1 - x + x² - x³ + ... = 1/(1+x) for |x| < 1

This infinite series representation offers another way to understand the function and its derivative. Differentiating the series term by term (a valid operation within the radius of convergence), we obtain:

-1 + 2x - 3x² + 4x³ - ...

This series represents the derivative of 1/(1+x) within its radius of convergence. This demonstrates the power of series representations in calculus.

Conclusion: A Multifaceted Problem with Broad Implications

Finding the derivative of 1/(1+x) may seem like a straightforward exercise, but it highlights the importance of various differentiation techniques, including the power rule, the chain rule, the quotient rule, and even implicit differentiation. Furthermore, its application in understanding the function's behavior, its connection to infinite series, and its role in various mathematical models underscores its significance in calculus and its applications in numerous scientific and engineering fields. This comprehensive analysis demonstrates how a seemingly simple problem can unveil a wealth of mathematical concepts and practical applications. Mastering these techniques provides a solid foundation for tackling more complex problems in calculus and beyond.