Finding The Second Derivative Of Y = X^(1/9) A Step-by-Step Guide

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In calculus, the second derivative of a function provides valuable information about the function's concavity and rate of change of its slope. In simpler terms, while the first derivative tells us how a function is changing, the second derivative tells us how the rate of change itself is changing. This is crucial in various applications, such as physics (analyzing acceleration), economics (understanding rates of economic growth), and optimization problems (finding maximum and minimum values).

This article will walk you through the process of finding the second derivative, denoted as d²y/dx², of the function y = x^(1/9). This function is a power function, and we'll be utilizing the power rule of differentiation repeatedly to arrive at our solution. Let's dive in!

Understanding Derivatives

Before we get into the specifics of this problem, let's briefly review what derivatives are and why they are so important.

The derivative of a function, often denoted as dy/dx or f'(x), represents the instantaneous rate of change of the function with respect to its independent variable (usually x). Geometrically, it gives the slope of the tangent line to the function's graph at a particular point. The derivative is a fundamental concept in calculus and has wide-ranging applications across various fields.

The second derivative, denoted as d²y/dx² or f''(x), is the derivative of the first derivative. In other words, it measures the rate of change of the slope of the original function. This tells us how the function's rate of change is changing. A positive second derivative indicates that the function is concave up (shaped like a U), while a negative second derivative indicates that the function is concave down (shaped like an upside-down U).

The second derivative is particularly useful in determining points of inflection, which are points where the concavity of the function changes. These points often represent significant changes in the behavior of the function and are crucial in optimization and curve sketching.

Problem Statement

We are given the function:

y = √[9]{x} = x^(1/9)

Our goal is to find the second derivative of this function, d²y/dx².

Step-by-Step Solution

1. Find the First Derivative (dy/dx)

To find the first derivative, we'll apply the power rule of differentiation. The power rule states that if y = x^n, then dy/dx = nx^(n-1). Applying this rule to our function, y = x^(1/9), we get:

dy/dx = (1/9)x^((1/9) - 1) dy/dx = (1/9)x^(-8/9)

So, the first derivative of y = x^(1/9) is dy/dx = (1/9)x^(-8/9).

2. Find the Second Derivative (d²y/dx²)

Now, we need to find the derivative of the first derivative, which is the second derivative. We'll again use the power rule. Let's differentiate dy/dx = (1/9)x^(-8/9) with respect to x:

d²y/dx² = d/dx [(1/9)x^(-8/9)] d²y/dx² = (1/9) * (-8/9)x^((-8/9) - 1) d²y/dx² = (-8/81)x^(-17/9)

Therefore, the second derivative of y = x^(1/9) is d²y/dx² = (-8/81)x^(-17/9).

3. Final Answer

The second derivative of the function y = x^(1/9) is:

d²y/dx² = (-8/81)x^(-17/9)

This is our final answer. We can also express this result using radicals if desired:

d²y/dx² = -8 / (81 * √[9]{x^17})

Deep Dive into the Concepts

Understanding the Power Rule: The power rule is a cornerstone of differential calculus. It provides a straightforward method for finding the derivative of power functions, which are functions of the form x^n, where n is a constant. The rule states that the derivative of x^n is nx^(n-1). This rule can be derived using the definition of the derivative and the binomial theorem, but it's often memorized and applied directly due to its frequent use.

Implications of the Second Derivative: The second derivative provides valuable information about the concavity of a function. Concavity describes the direction in which a curve bends. A positive second derivative indicates that the function is concave up (shaped like a U), meaning the rate of change of the slope is increasing. Conversely, a negative second derivative indicates that the function is concave down (shaped like an upside-down U), meaning the rate of change of the slope is decreasing.

Points where the second derivative is zero or undefined are potential inflection points. An inflection point is a point on the curve where the concavity changes. To confirm that a point is an inflection point, the second derivative must change sign at that point.

Applications in Curve Sketching: The first and second derivatives are essential tools in curve sketching. The first derivative helps identify critical points (local maxima and minima) and intervals where the function is increasing or decreasing. The second derivative helps determine the concavity of the function and locate inflection points. By combining this information, we can create a detailed sketch of the function's graph.

Common Mistakes and How to Avoid Them

When finding derivatives, especially higher-order derivatives, it's easy to make mistakes. Here are some common pitfalls and strategies to avoid them:

  1. Incorrectly Applying the Power Rule: The power rule is straightforward, but it's crucial to apply it correctly. Ensure you multiply by the exponent and then subtract 1 from the exponent. For example, the derivative of x^(1/2) is (1/2)x^(-1/2), not (1/2)x^(1/2).

  2. Forgetting the Chain Rule: While this problem didn't require the chain rule, it's essential to remember it for more complex functions. If you're differentiating a composite function (a function within a function), you need to apply the chain rule. The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).

  3. Algebraic Errors: Simplifying expressions with negative or fractional exponents can be tricky. Double-check your algebraic manipulations to avoid errors. For example, ensure you correctly simplify expressions like x^(-8/9) * x^(-1).

  4. Confusing First and Second Derivatives: Remember that the second derivative is the derivative of the first derivative. Don't accidentally differentiate the original function twice. Clearly label your steps to avoid this confusion.

  5. Not Simplifying the Result: While the unsimplified derivative is technically correct, it's often helpful to simplify the expression. This can make it easier to analyze the function's behavior and use the derivative in further calculations.

Practice Problems

To solidify your understanding of finding second derivatives, try these practice problems:

  1. Find d²y/dx² for y = x^5 - 3x^3 + 2x.
  2. Find d²y/dx² for y = 1/x^2.
  3. Find d²y/dx² for y = √(x^3).

Work through these problems step-by-step, paying attention to the power rule and algebraic simplification. Check your answers by comparing them to solutions available online or in calculus textbooks.

Conclusion

In this article, we successfully found the second derivative of the function y = x^(1/9). We applied the power rule of differentiation twice to arrive at the solution d²y/dx² = (-8/81)x^(-17/9). Understanding how to find second derivatives is crucial for analyzing the concavity of functions and solving various problems in calculus and its applications. By practicing and paying attention to common mistakes, you can master this essential skill.

Remember, the journey to calculus mastery is paved with practice and perseverance. Keep exploring, keep learning, and you'll continue to deepen your understanding of this powerful mathematical tool.