Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ h(x)=\frac{x}{x-5} $$

Step-by-Step Solution

Answer

The derivative of the function \(h(x) = \frac{x}{x-5} \) is \(h'(x) = \frac{-5}{(x-5)^2}\). The Quotient Rule for differentiation was used to find this derivative.

Step 1: Apply the Quotient Rule

The quotient rule for differentiation states that for any two differentiable functions, \(u(x)\) and \(v(x)\), the derivative of their quotient, given by \(w = \frac{u(x)}{v(x)}\), is: \(w' = \frac{u'v - uv'}{v^2}\). We identify \(u(x) = x\) and \(v(x) = x - 5\). So, we calculate \(u' = 1\) and \(v' = 1\). Now we substitute \(u, u', v, v'\) into the quotient rule to get the derivative of \(h\).

Step 2: Calculate the derivative

The derivative \(h'(x) = \frac{u'v - uv'}{v^2} = \frac{(1)(x-5) - (x)(1)}{(x-5)^2} = \frac{x-5 - x}{(x-5)^2} = \frac{-5}{(x-5)^2}\). Thus, the derivative of the function \(h(x) = \frac{x}{x-5} \) is \(h'(x) = \frac{-5}{(x-5)^2}\).

Step 3: Provide the Differentiation Rule Used

The differentiation rule that was used to find the derivative of the function is the Quotient Rule, which is used when a function is the quotient of two other functions.

Key Concepts

Derivative of a Quotient

Understanding the derivative of a quotient is crucial for calculus students as it's a common operation in many problems. The quotient rule is a formal method to find the derivative when a function is written as a fraction where both the numerator and the denominator are themselves functions of the variable.

Let's consider a function in the form of a quotient, where the numerator is a function labeled as u(x) and the denominator as v(x), represented by:
\[ h(x) = \frac{u(x)}{v(x)} \].

The derivative, or the rate of change, of this function according to the quotient rule is described by the formula:
\[ h'(x) = \frac{u'(x) \times v(x) - u(x) \times v'(x)}{v(x)^2} \].
Here, u'(x) and v'(x) are the derivatives of u(x) and v(x), respectively. By applying this rule, the complexity of differentiating a quotient is reduced to a series of simpler differentiation and algebraic steps.

Differentiable Functions

A fundamental prerequisite for applying the quotient rule is that both functions in the numerator and denominator, u(x) and v(x), must be differentiable functions. A function is said to be differentiable at a point if it has a derivative there, implying a smooth and continuous behavior at that point.

For a function to be differentiable across an interval, it must not only be continuous but also it must not have any sharp corners or cusps. In the context of our function h(x), we ensure that the original functions making up the numerator and denominator are differentiable before applying the quotient rule. The existence of this derivative allows us to compute the behavior of the function's rate of change at any given point.

Calculating Derivatives

The process of calculating derivatives is at the heart of differential calculus. The derivative of a function at a point characterizes the rate of change of the function at that point. Various rules exist for calculating derivatives, depending on the form of the function. For polynomial functions, power rules are used; for exponential functions, specific exponential rules are applied.

In the case of our example with the function h(x), we use the quotient rule for calculating the derivative because the function is a ratio of two polynomials. By identifying the correct rule to use, we expedite the calculation and can focus on the algebraic manipulation to simplify the resulting expression.

Differentiation Rules

Differential calculus is built on a set of well-defined differentiation rules. Knowledge of these rules enables students to approach the calculation of derivatives systematically. Rules such as the power rule, product rule, quotient rule, and chain rule, each have their own unique conditions of use and result in different derivative expressions based on the functions they are applied to.

In the provided exercise, the quotient rule was utilized to determine the derivative of the function. The quotient rule is specifically chosen when a function is represented as a division of two functions, as seen with our function h(x). Remembering these differentiation rules and knowing when to apply them will significantly enhance the ability to tackle calculus problems efficiently and accurately.