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Magazine Chapter 1: Problem 12
Find \(d^{2} y / d x^{2}\) $$ y=\sqrt[4]{x} $$
Short Answer
Expert verified
The second derivative is \( \frac{d^2y}{dx^2} = -\frac{3}{16x^{7/4}} \).
Step by step solution
01
Differentiate the Original Function
Given the function \( y = \sqrt[4]{x} \), rewrite it as \( y = x^{1/4} \). Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \). Using the power rule, \( \frac{d}{dx}[x^n] = nx^{n-1} \), we get: \[ \frac{dy}{dx} = \frac{1}{4}x^{-3/4} \]
02
Simplify the First Derivative
The expression for the first derivative is \( \frac{dy}{dx} = \frac{1}{4}x^{-3/4} \). Simplifying, we can rewrite it in the form \( \frac{1}{4}\left(\frac{1}{x^{3/4}}\right) \) or \( \frac{1}{4x^{3/4}} \).
03
Differentiate Again to Find the Second Derivative
Differentiate \( \frac{dy}{dx} = \frac{1}{4}x^{-3/4} \) again with respect to \( x \) to find \( \frac{d^2y}{dx^2} \). Using the power rule again: \[ \frac{d^2y}{dx^2} = \frac{1}{4} \times (-3/4) x^{-7/4} \] Simplifying, we obtain: \[ \frac{d^2y}{dx^2} = -\frac{3}{16}x^{-7/4} \]
04
Simplify the Second Derivative
Expressing \( \frac{d^2y}{dx^2} = -\frac{3}{16}x^{-7/4} \) in a more readable form gives: \[ \frac{d^2y}{dx^2} = -\frac{3}{16} \frac{1}{x^{7/4}} \] or \[ \frac{d^2y}{dx^2} = -\frac{3}{16x^{7/4}} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Power Rule
The power rule is a fundamental principle in calculus that simplifies the process of differentiation. When you are tasked with differentiating a function of the form \( x^n \), where \( n \) is any real number, the power rule provides a simple method to find its derivative. This rule states that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \).
- For example, if \( n = 4 \), the derivative would be \( 4x^{3} \).
- When \( n = 1/4 \), as in our problem, the derivative becomes \( \frac{1}{4}x^{-3/4} \).
The Process of Differentiation
Differentiation is the mathematical process through which we determine the rate at which a quantity changes with respect to another. This is typically represented as the derivative of a function.
- In calculus, differentiation provides insights into the behavior of functions, describing how they increase, decrease, and where they might reach local maxima or minima.
- When we differentiate a function like \( y = x^{1/4} \), we aim to find \( \frac{dy}{dx} \), which gives us the slope of the tangent line to the curve at any point \( x \).
Exploring Functions and Their Characteristics
In mathematics, functions are expressions that define a specific relationship between two variables, typically \( x \) and \( y \). Each input (\( x \)) has a corresponding output (\( y \)). Functions can represent a wide range of real-world phenomena, from the trajectory of a thrown ball to the growth of a population.
- Functions can be linear, quadratic, polynomial, exponential, or various other types, each with their own properties and behaviors.
- For instance, the function \( y = \sqrt[4]{x} \) is a non-linear function where inputs are roots, meaning each value of \( x \) is transformed by taking its fourth root. This gives a unique shape to the graph of the function.
