Problem 1 - Thomas Calculus

Question

Find \(d y / d x\) in Exercises \(1-10\) $$ y=x^{9 / 4} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = \frac{9}{4}x^{5/4} \).

Step 1: Understand the Problem

The problem asks us to find the derivative of the function \( y = x^{9/4} \) with respect to \( x \).

Step 2: Apply the Power Rule

The power rule for differentiation states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \). In this problem, \( n = \frac{9}{4} \).

Step 3: Differentiate the Function

Using the power rule, differentiate \( y = x^{9/4} \). This gives us \( \frac{dy}{dx} = \frac{9}{4}x^{(9/4)-1} \).

Step 4: Simplify the Expression

Simplify the expression \( \frac{dy}{dx} = \frac{9}{4}x^{5/4} \) to ensure it is in the simplest form.

Key Concepts

Power Rule

The power rule is a straightforward and essential tool for budding calculus students. It provides a quick way to differentiate functions of the form \(y = x^n\). Essentially, if you want to find the derivative of \(x^n\) with respect to \(x\), the power rule states that you simply bring down the exponent \(n\) as a coefficient and then reduce the exponent by 1. So it transforms to \( \frac{dy}{dx} = nx^{n-1} \). This means:

• Take the power (exponent) in the original function.
• Multiply the entire term by that power.
• Reduce the original power by 1.

This rule is particularly useful because it saves time and effort in calculations over more complex methods. In our specific example, when \( y = x^{9/4} \), the power rule has helped us to quickly determine that \( \frac{dy}{dx} = \frac{9}{4}x^{5/4} \). This shows the efficiency and simplicity of using the power rule for standard polynomials or terms involving a single exponent.

Differentiation

Differentiation is the process of finding a derivative, which is essentially the rate at which something changes. In mathematics, particularly in calculus, differentiation is used to determine how a function's value changes as its input changes.

• The derivative of a function gives the slope of the tangent line at any point on the curve.
• It's a fundamental concept that allows us to understand how variables are related to one another.
• In simple terms, the derivative tells us how fast or slow the change is happening.

In the context of this exercise, differentiation involved applying the power rule to the function \(y = x^{9/4}\). By differentiating, we established that the rate of change of \(y\) concerning \(x\) is \(\frac{9}{4}x^{5/4}\). This gives us insights into the behavior of the function as \(x\) varies.

Calculus

Calculus is a branch of mathematics that explores how things change. It is divided into two main areas: differentiation and integration. For now, we're focusing on differentiation, which is concerned with understanding rates of change and slopes of curves.Calculus finds applications in various fields, making it invaluable:

• Physics uses calculus to describe the motion and dynamics of objects.
• Economics applies it to optimize functions like cost, revenue, and profit.
• Biology leverages calculus to model population growth and decay.

Our exercise falls under the differentiation aspect of calculus, where we took the function \(y = x^{9/4}\) and applied calculus principles to find its derivative. This process allows us to understand the continuous change in the function’s output based on its input, building a foundational understanding crucial for more advanced studies in calculus and related fields.