Question

Finding a Second Derivative In Exercises \(91-98\) , find the second derivative of the function. $$ f(x)=\frac{x}{x-1} $$

Short answer

The second derivative of the function \(f(x)=\frac{x}{x-1}\) is \(f''(x) = -2 / (x - 1)^3\).

Step-by-step solution

Find the First Derivative

To find the first derivative, apply the quotient rule. The quotient rule is \((u / v)' = (v*du - u*dv) / v^2\) where 'u' is the numerator (which here is 'x'), 'v' is the denominator (which here is \(x - 1\)), 'du' is the derivative of 'u', and 'dv' is the derivative of 'v'. Using the quotient rule, the first derivative \(f'(x)\) is given by \((x - 1 - x) / (x - 1)^2 = -1 / (x - 1)^2\).

Find the Second Derivative

Now, use the derivative found in step 1 to calculate the second derivative \(f''(x)\). Again, apply the quotient rule with 'u' as the numerator (which here is '-1'), 'v' as the denominator (which here is \(x - 1^2\)), 'du' is the derivative of 'u', and 'dv' is the derivative of 'v'. Using the quotient rule, the second derivative \(f''(x)\) is given by \((2(x-1)*(-1)) / (x - 1)^4 = -2 / (x - 1)^3\).

Key concepts

Calculus

Calculus is an essential aspect of mathematics, often focusing on concepts such as limits, derivatives, and integrals. It helps us understand how things change over time or space. In calculus, one of the primary objectives is to find the rate of change of a function. This is achieved through differentiation. By understanding the rate at which variables affect outcomes, calculus provides insight into motion, growth, and trends.

• Calculus allows us to model and analyze real-world phenomena.
• It encompasses both differential calculus and integral calculus.
• Derivatives and integrals serve as the foundation of calculus operations.

It is widely used in fields such as physics, engineering, economics, and statistics. Having a grasp on calculus is instrumental in solving complex problems with multiple variables.

Quotient Rule

The quotient rule is an essential technique in calculus, specifically in the differentiation process. It is used when differentiating a function expressed as a fraction of two other functions, known as the numerator and denominator. The rule formulates how to find the derivative of such a quotient, and is written as:\[ (\frac{u}{v})' = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]where:

• \( u \) is the numerator
• \( v \) is the denominator
• \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) are the derivatives of \( u \) and \( v \) respectively.

To use the quotient rule effectively, follow these steps:

• Differentiate the numerator (\( u \)) and the denominator (\( v \)).
• Substitute their derivatives back into the formula.
• Simplify to find the final expression for the derivative.

The quotient rule is crucial when dealing with rational functions, providing a systematic approach to finding their derivatives.

Differentiation

Differentiation is a core process within calculus, designed to determine the rate of change of a function. This process is fundamental to finding derivatives, which represent the slope of a function at any given point. In essence, differentiation allows us to understand how a function behaves as its input varies.

• Differentiation provides the gradient or steepness of the curve of a function.
• It helps in determining local maxima and minima of functions.
• The process of differentiation can be applied repeatedly to acquire higher-order derivatives like the second derivative.

To differentiate simple functions, rules such as the power rule, product rule, and quotient rule are employed. Efficient differentiation aids in analyzing the functional relationship between variables across various contexts, from physics to finance.

First Derivative

The first derivative of a function is the initial step in examining how the function changes. It tells us the slope of the tangent line to the curve of the function at any given point. In mathematical terms, the first derivative can indicate whether a function is increasing or decreasing at a specific point.To find the first derivative:

• Identify the function's rule, such as applying the power rule or quotient rule.
• Calculate the derivative using these established rules or techniques.
• Simplify the derivative expression to understand the behavior of the function.

In our exercise, we started by calculating the first derivative of the function \( f(x) = \frac{x}{x-1} \) using the quotient rule, resulting in \( f'(x) = -\frac{1}{(x-1)^2} \).
The first derivative plays a significant role in determining the function's behavior, predicting points of inflection, and serving as a step towards calculating higher-order derivatives like the second derivative.