Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=\frac{x+4}{4-x}$$

Short answer

Question: Given the function f(x) = (x+4)/(4-x), find the relationship between a small change in x (dx) and the corresponding change in y (dy) in the form dy = f'(x)dx. Answer: The relationship between a small change in x (dx) and the corresponding change in y (dy) for the given function f(x) = (x+4)/(4-x) is: dy = (-2x/(4-x)^2)dx.

Step-by-step solution

Identify the given function

The given function is: $$f(x)=\frac{x+4}{4-x}$$ Our goal is to find the relationship between a small change in x (dx) and the corresponding change in y (dy), expressed in the form dy = f'(x)dx.

Find the derivative of the function f(x)

First, we need to find the derivative of f(x) with respect to x: To differentiate $$f(x)=\frac{x+4}{4-x}$$, we will use the quotient rule which states that for any functions u(x) and v(x), $$ \frac{d}{d x}\left(\frac{u(x)}{v(x)}\right)=\frac{u^{\prime}(x) v(x)-u(x) v^{\prime}(x)}{v^{2}(x)} $$So, let \(u(x) = x + 4\) and \(v(x) = 4 - x\). The derivatives of u(x) and v(x) with respect to x are: $$u^{\prime}(x)=\frac{d}{d x}(x+4)=1$$ $$v^{\prime}(x)=\frac{d}{d x}(4-x)=-1$$Now, applying the quotient rule, we find the derivative of f(x): $$f^{\prime}(x)=\frac{(1)(4-x)-(x+4)(-1)}{(4-x)^{2}}$$ Now, simplify the expression: $$f^{\prime}(x)=\frac{4-x+(-x-4)}{(4-x)^{2}}$$ $$f^{\prime}(x)=\frac{-2x}{(4-x)^{2}}$$ So, the derivative of $$f(x) = \frac{x + 4}{4 - x}$$ is $$f^{\prime}(x) = \frac{-2x}{(4-x)^{2}}$$.

Express the relationship between dy and dx using the derivative

We found the derivative f'(x) = $$\frac{-2x}{(4-x)^{2}}$$, so we can write the relationship between a small change in x (dx) and the corresponding change in y (dy) as: $$d y=f^{\prime}(x) d x=\frac{-2x}{(4-x)^{2}}d x$$ This expression shows how a small change in x will result in the corresponding change in y, in the form dy = f'(x)dx for the given function f(x).

Key concepts

Derivative of a Function

The derivative of a function is like the function's secret tool for measuring how it changes. Imagine a function as a machine. When you put a little value like \(x\) into it, it gives you an output, say \(y\). The derivative, denoted as \(f'(x)\), tells us how sensitive the output \(y\) is to changes in the input \(x\). In simple terms, the derivative can tell us how fast \(y\) is changing at any given point with respect to \(x\).

In calculus, we use derivatives to understand rates of change, much like speedometers measure speed. For example, if we have a function \(f(x) = \frac{x+4}{4-x}\), finding \(f'(x)\) involves differentiating the function with respect to \(x\). This process is critical because knowing \(f'(x)\) helps to predict the behavior of the function when \(x\) changes, such as in physics when calculating velocity from position.

Small Change in Variables

In mathematics, a small change in variables reflects how a slight adjustment in one quantity affects another quantity. When you slightly change \(x\), it results in a small change in \(y\). This notion is expressed in an equation as \(dy = f'(x)dx\), where \(dy\) is the small change in \(y\) and \(dx\) is the small change in \(x\).

Think of \(dx\) as taking a tiny step on the \(x\)-axis. This step causes \(y\) to change by \(dy\), which is predicted by multiplying \(dx\) by the derivative \(f'(x)\). Essentially, this relationship helps us calculate how slight movements in one direction affect movements in another direction.

• This concept is crucial for approximations in engineering and sciences, where exact values are hard to obtain but small changes can provide significant insights.
• Understanding this relationship also helps in curve sketching and optimization problems, where knowing how one variable subtly impacts another is essential.

Quotient Rule

The quotient rule is a helpful technique in calculus used to differentiate functions presented as a division of two other functions. When faced with a function \(f(x) = \frac{u(x)}{v(x)}\), the quotient rule provides a systematic way to find \(f'(x)\), the derivative of \(f(x)\).

The rule states that for functions \(u(x)\) and \(v(x)\), the derivative \(\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)\) is given by:\[\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\] Here, \(u'(x)\) and \(v'(x)\) are the derivatives of \(u(x)\) and \(v(x)\), respectively.

• This formula helps us calculate the derivative of a function that looks complicated due to division.
• For example, with \(f(x) = \frac{x+4}{4-x}\), the quotient rule allows us to systematically find \(f'(x)\).
• Using the quotient rule ensures accuracy, as it carefully accounts for both functions \(u(x)\) and \(v(x)\) and their rates of change.