Question

If \(y\) is related to \(x\) by \(y=x^{k},\) how will the rates of growth \(r_{y}\) and \(r_{x},\) be related?

Short answer

\( r_y = kx^{k-1} \cdot r_x \): The rate of growth of \( y \) is \( kx^{k-1} \) times the rate of growth of \( x \).

Step-by-step solution

Understanding the Relationship

We are given that the relationship between \(y\) and \(x\) is defined by \(y = x^k\). This implies that \(y\) is a power of \(x\). Here, \(k\) is a constant exponent, and we are interested in finding how the rates of growth of \(y\) (denoted as \(r_y\)) and the rates of growth of \(x\) (denoted as \(r_x\)) relate to each other.

Define Rates of Growth

The rate of growth \( r_x \) of \( x \) is defined as the derivative \(\frac{dx}{dt}\), while the rate of growth \( r_y \) of \( y \) is defined as \(\frac{dy}{dt}\). We need to establish how \( \frac{dy}{dt} \) relates to \( \frac{dx}{dt} \).

Differentiate the Given Function

To find \( \frac{dy}{dt} \), we differentiate \( y = x^k \) with respect to \( t \).The chain rule gives:\[\frac{dy}{dt} = \frac{d}{dt}(x^k) = kx^{k-1} \cdot \frac{dx}{dt}\]

Express the Relationship Between Rates

From the expression derived in the previous step, we can write the rate of growth of \( y \) in terms of the rate of growth of \( x \):\[ \frac{dy}{dt} = kx^{k-1} \cdot \frac{dx}{dt}\]This shows that \( r_y = kx^{k-1} \cdot r_x \). Thus, the rate of growth of \( y \) is proportional to the rate of growth of \( x \) multiplied by \( kx^{k-1} \).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus. It allows us to differentiate composite functions. In our exercise, we have the function relationship \( y = x^k \). This is a simple case where we treat \( y \) as a function of \( x \), and \( x \) as a function of time \( t \).

The chain rule helps us find the derivative of a function that is nested within another function. The principle can be phrased as: "To differentiate the outer function, multiply the derivative of the outer function by the derivative of the inner function."

• First, identify the outer function; here, it's \( x^k \).
• Next, differentiate \( x^k \) with respect to \( x \), using power rule, making it \( kx^{k-1} \).
• Then, multiply this by the derivative of \( x \) with respect to \( t \), yielding the composition \( kx^{k-1} \cdot \frac{dx}{dt} \).

This process provides us a powerful tool for finding rates of change in many scientific and engineering contexts, giving us insight into how changes in one quantity affect another.

Power Function

Power functions like \( y = x^k \) are mathematical expressions where a variable \( x \) is raised to a constant power \( k \). Understanding how to differentiate these functions is crucial, as they commonly appear in polynomial equations and many physical laws.

To find the derivative of a power function, we use the "Power Rule," a cornerstone of derivative calculus. This rule states that for \( y = x^n \), its derivative with respect to \( x \) is \( nx^{n-1} \).

In practice, this means:

• If \( y = x^k \), then \( \frac{d}{dx}(x^k) = kx^{k-1} \).
• This helps us determine the rate at which \( y \) changes with \( x \).

The power rule simplifies the calculation of derivatives for exponential growth and decay functions, making it an essential tool in calculus.

Rates of Growth

Rates of growth refer to how quickly one quantity changes in response to a change in another quantity. In our context, we have \( r_y\) as the rate of growth of \( y \) and \( r_x \) as the rate of growth of \( x \).

The main task is to relate \( r_y = \frac{dy}{dt} \) to \( r_x = \frac{dx}{dt} \). Using differentiation, specifically the chain rule applied to a power function, we found that:

• \( \frac{dy}{dt} = kx^{k-1} \cdot \frac{dx}{dt} \).
• Thus, \( r_y = k x^{k-1} \times r_x \).

This equation reveals the proportionality between the rates of growth. The rate \( r_y \) depends on both the rate \( r_x \) and the expression \( kx^{k-1} \), representing how steep the curve of the function is at any point \( x \). Such relationships are widely used in mathematics and science to model dynamic systems.