Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\). \(f(x)=2 x+1\)

Short answer

Answer: The relationship between a small change in \(x\) and the corresponding change in \(y\) for the function \(f(x) = 2x + 1\) is expressed as \(dy = 2 dx\).

Step-by-step solution

Differentiating the Function

To differentiate \(f(x) = 2x + 1\) with respect to \(x\), we apply the power rule. The power rule states that if \(f(x) = ax^n\), then \(f'(x) = nax^{n-1}\). In our case, we have two terms: \(2x^1\) and \(1x^0\). Let's differentiate each term with respect to \(x\). For \(2x^1\): \(f'(x) = 1(2x^{1-1}) = 2x^0 = 2\) For \(1x^0\): Since the term is a constant, the derivative is 0. Now, combine the results to get the derivative: \(f'(x) = 2 + 0 = 2\).

Expressing the Relationship

Now that we have determined the derivative \(f'(x) = 2\), we can express the relationship between the small change in \(x\) and the corresponding change in \(y\) using the formula \(dy = f'(x) dx\). Replace \(f'(x)\) with the value we found in Step 1: \(dy = 2 dx\) Thus, the relationship between a small change in \(x\) and the corresponding change in \(y\) for the function \(f(x) = 2x + 1\) is expressed as \(dy = 2 dx\).

Key concepts

Differentiation

Differentiation is a cornerstone concept in mathematics, particularly in differential calculus. It involves finding the derivative of a function, which is synonymous with discovering the rate at which the function's output value changes with respect to a change in its input value. To visualize this better, imagine you're driving a car on a straight road; differentiation would be like looking at the speedometer to find out how fast your speed is changing at any given moment.

Differentiation is crucial because it allows us to understand how slight changes in one variable, like time or distance, affect another, such as speed or position. This understanding’s implications are vast, stretching from simple motion to complex changes in finance, physics, engineering, and beyond.

Power Rule

The power rule is a quick and handy tool in differentiation for dealing with polynomials, where the exponent is a real number. The rule says that if you have a function of the form \(f(x) = ax^n\), where \(a\) is a constant and \(n\) is a real number, the derivative of that function with respect to \(x\) is \(f'(x) = nax^{n-1}\).

For example, when you have \(x^3\), applying the power rule gives you \(3x^2\) as the derivative. The power rule simplifies the differentiation process; no need for complex computations or limits, just a straightforward application of the rule. It's often one of the first techniques taught to students learning calculus due to its efficiency and utility in solving various problems.

Derivative

The derivative of a function represents the slope of the tangent to the function's graph at any point or the function's instantaneous rate of change. The derivative is fundamental to calculus and is a tool that measures how a small change in one variable, often \(x\), causes a change in another variable, usually depicted as \(y\) or \(f(x)\). A function's derivative is frequently denoted as \(f'(x)\) or \(\frac{dy}{dx}\) when represented as a ratio. Understanding the derivative is key to navigating complex equations, predicting patterns, and solving real-world problems like optimizing resources or modeling population growth.

Small Change in x and y

In the context of differential calculus, a 'small change' in variables \(x\) and \(y\) is often represented using the symbols \(dx\) and \(dy\). These notations symbolize infinitesimal increments or decrements in the values of \(x\) and \(y\). The key to understanding this concept is to grasp that calculus is the study of continuous change, and these 'small changes' are so minute that they approach zero without ever getting there.

When a function \(y = f(x)\) is differentiated, \(dy\) represents the change in \(y\) that corresponds to an infinitesimally small change in \(x\), denoted as \(dx\). The equation representing this relationship is \(dy = f'(x) dx\), and it is fundamental for problems involving rate of change or slope. This concept intertwines with the derivative, extending one's ability to analyze motion, change, and trends in a variety of disciplines.