Problem 26 - Calculus and its applications

Question

Find \(y^{\prime \prime}\) $$ y=x^{2 / 3}+4 x $$

Step-by-Step Solution

Answer

The second derivative is \( y'' = \frac{-2}{9}x^{-4/3} \).

Step 1: Understand the Function

The given function is \( y = x^{2/3} + 4x \). Our task is to find the second derivative of this function, which is denoted as \( y'' \).

Step 2: Find the First Derivative

To find \( y'' \), we first need to find the first derivative \( y' \). Derive the function \( y = x^{2/3} + 4x \). The derivative of \( x^{2/3} \) is \( \frac{2}{3}x^{-1/3} \) and the derivative of \( 4x \) is \( 4 \). Combining these, we get: \[ y' = \frac{2}{3}x^{-1/3} + 4 \]

Step 3: Find the Second Derivative

Differentiate \( y' \) to find the second derivative \( y'' \). The derivative of \( \frac{2}{3}x^{-1/3} \) is \( \frac{-2}{9}x^{-4/3} \) and the derivative of a constant, 4, is 0. Thus, we have:\[ y'' = \frac{-2}{9}x^{-4/3} \]

Step 4: Write the Final Answer

The second derivative of the given function is:\[ y'' = \frac{-2}{9}x^{-4/3} \]

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on studying the rates of change. One of the fundamental tools of calculus is differentiation. This is a technique that allows us to determine how a function changes at any point. Calculus helps us understand curves, areas under curves, and how different quantities interact in dynamic ways. In physics, for example, calculus is crucial for analyzing motion and forces.

• Calculus breaks down into two major subfields: differential calculus and integral calculus.
• Differential calculus is about finding the derivative, which measures how a function's output changes as its input changes.
  
• Integral calculus, on the other hand, focuses on the area under the curve and accumulative quantities.
  

By using calculus, we can solve complex problems involving motion, growth, and decay in various functions. The second derivative, in particular, provides insights into the rate at which the rate of change is changing, offering a deeper understanding of the behavior of the function.

Differentiation

Differentiation is the mathematical process of finding a derivative. It involves rules and techniques that help you find the rate at which a function changes. In simpler terms, it tells us how fast something is happening, whether it is speed, population growth, or something else.
When we differentiate, we're looking at the slope of the tangent line to the curve at any point. A key purpose of differentiation is to find maxima and minima in functions, which are the highest and lowest points in a curve.

• The first derivative of a function is commonly denoted as \( y' \) or \( f'(x) \). It describes the rate of change itself.
• The second derivative, \( y'' \) or \( f''(x) \), indicates the concavity of the function. A positive second derivative suggests the function is concave up, while a negative one suggests it's concave down.
• Higher-order derivatives can provide even more insights into the function's characteristics.
  

Differentiation helps predict future values and analyze trends, making it an important tool in science and engineering.

Mathematical Functions

A mathematical function defines a relationship between a set of inputs and their corresponding outputs. Each input in a function maps exactly to one output, following specific rules.
Functions can be as simple as linear equations like \( y = 4x + 2 \) or as complex as polynomial or trigonometric functions. Understanding mathematical functions is crucial because they can describe real-life situations, such as financial models or physical phenomena.

• Functions are typically denoted by letters like \( f(x) \), \( g(x) \), providing a formula or rule for how the input \( x \) is processed.
• The graph of a function gives a visual representation of this relationship, showing how the output changes with the input.
• Some functions are continuous, meaning they have no breaks, while others might have restrictions leading to gaps or jumps in their graphs.
  

With functions, we can explore and describe changes in variables systematically. This is foundational knowledge in calculus, as it allows us to apply differentiation techniques to understand deeper properties of these relationships.