Problem 4 - Basic Engineering Mathematics

Question

Find the differential coefficient of \(y=5 x\)

Step-by-Step Solution

Answer

The differential coefficient of \( y=5x \) is 5.

Step 1: Identify the Function

The given function is a simple linear function, expressed as \( y = 5x \). This represents a straight line with a constant slope.

Step 2: Use the Power Rule

The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Here, the function \( y = 5x \) can be rewritten as \( y = 5x^1 \).

Step 3: Apply the Constant Rule

Since 5 is a constant coefficient, it can be factored out of the differentiation process. Therefore, \( \frac{d}{dx}(5x) = 5 \cdot \frac{d}{dx}(x) \).

Step 4: Differentiate the Power Function

Differentiate \( x \) using the power rule. The derivative of \( x^1 \) is \( 1x^{1-1} = 1 \).

Step 5: Calculate the Differential Coefficient

Multiply the constant by the result of the differentiation: \( 5 \times 1 = 5 \). The differential coefficient of \( y = 5x \) is 5.

Key Concepts

Differentiation

Differentiation is an essential concept in calculus that deals with finding the derivative of a function. It's a way to determine how a function changes at any given point.
Put simply, a derivative tells you the slope of a curve at a particular point, allowing you to understand the rate of change. When we talk about differentiation, there are some basic rules and methods that come into play.

• Derivatives: A derivative is the result of differentiation, expressed in a way that shows how one quantity changes as another changes.
• Notation: We often denote derivatives with symbols like \( f'(x) \), \( \frac{dy}{dx} \), or \( Df(x) \).
• Purpose: Differentiation has applications in various fields such as physics, engineering, and economics, offering insights into behaviors like velocity, growth rates, and trends.

Once you grasp the basics of differentiation, you can tackle more complex problems by applying specific rules like the power rule.

Power Rule

The power rule is a fundamental tool in calculus for differentiating polynomial functions. It greatly simplifies the process when dealing with functions where a variable is raised to a power.
Here's the formal statement: if you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \). This rule is straightforward to use but extremely powerful.

• Explanation: The power rule lets you quickly find the rate of change of any power function.
• Application: Take \( y = 5x \) as an example. This can be rewritten as \( y = 5x^1 \) to apply the power rule more easily.
• Steps: To differentiate \( x^1 \), apply the rule: multiply the exponent by the coefficient, then decrease the exponent by one. Here, \( \frac{d}{dx}(x^1) = 1 \cdot x^{0} = 1 \).

Using the power rule effectively reduces the time spent on differentiation tasks, especially for polynomial expressions.

Linear Function

A linear function is one of the simplest types of functions in mathematics, characterized by its constant rate of change. It is usually expressed in the form \( y = mx + b \), where \( m \) signifies the slope and \( b \) the y-intercept.

• Slope: The slope \( m \) indicates how much \( y \) increases or decreases as \( x \) increases by one unit. For the function \( y = 5x \), the slope is 5, showing a linear relationship where \( y \) increases five times as fast as \( x \).
• Graph: When plotted on a graph, a linear function forms a straight line, illustrating a uniform rate of change.
• Characteristics: Linear functions have no curvature; the derivative (or slope) is constant across all points.

Understand that linear functions are foundational in calculus, simplifying the differentiation process as seen with the derivative of \( y = 5x \) being a straightforward result.