Question

Finding a Second Derivative In Exercises \(91-98\) , find the second derivative of the function. $$ f(x)=\frac{x^{2}+3 x}{x-4} $$

Short answer

\[f''(x) = \frac{-2x^2 + 7x + 20}{(x-4)^3}\]

Step-by-step solution

Apply quotient rule to find the first derivative

Let \(u = x^2+3x\) and \(v = x-4\). The derivatives \(u'\) and \(v'\) are \(2x+3\) and \(1\) respectively. According to the quotient rule, \(f'(x) = \frac{(2x+3)(x-4) - (x^2+3x)(1)}{(x-4)^2}\.

Simplify the first derivative

By expanding the numerator and simplifying, we get \(f'(x) = \frac{2x^2 - 5x - 12}{(x-4)^2}\).

Apply quotient rule to find the second derivative

In order to find the second derivative, we need to differentiate \(f'(x)\) again. Viewing \(u = 2x^2 - 5x - 12\) and \(v = x-4\), and using the quotient rule, \(f''(x) = \frac{(4x-5)(x-4) - (2x^2 - 5x - 12)(1)}{(x-4)^3}.\)

Simplify the second derivative

By expanding the numerator and simplifying, we get \(f''(x) = \frac{-2x^2 + 7x + 20}{(x-4)^3}\). This is the second derivative of the function.

Key concepts

Quotient Rule

The Quotient Rule is a method used in calculus to find the derivative of a function that is expressed as the division (") of two differentiable functions. If you have a function described as \( f(x) = \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), then the derivative \( f'(x) \) will be given by the formula:

• \( f'(x) = \frac{u'v - uv'}{v^2} \)

To successfully apply the quotient rule, follow these steps:

• Identify the functions \( u \) and \( v \) and their respective derivatives \( u' \) and \( v' \).
• Plug these into the quotient rule formula.
• Simplify the expression by carrying out the necessary algebraic operations.

The Quotient Rule is particularly useful for differentiating rational functions, such as the one given in the problem \( f(x) = \frac{x^2 + 3x}{x-4} \). It allows us to calculate not just the first derivative, but is foundational in determining higher-order derivatives like the second derivative, as shown in this exercise.

Differentiation

Differentiation is the process used in calculus to determine the derivative of a function. The derivative represents the rate at which a function is changing at any given point and is a fundamental concept for understanding motion, rates, and function behaviors. To differentiate a function:

• Understand the rules of differentiation, such as the power rule, product rule, and quotient rule.
• Apply the corresponding rule based on how the function is structured.
• Simplify the resulting expressions to obtain the derivative function.

In our problem, the function is expressed as a fraction, necessitating the use of the quotient rule for differentiation. The first derivative \( f'(x) \), reflects how the original function \( f(x) = \frac{x^2 + 3x}{x-4} \) changes with respect to \( x \). By differentiating \( f'(x) \) once again, we find \( f''(x) \), the second derivative, which further informs us about the concavity and inflection points of \( f(x) \). Differentiation, therefore, is crucial for analyzing a function's overall behavior.

Simplifying Derivatives

Simplifying derivatives is an essential skill in calculus, especially after applying rules like the quotient rule. Once you determine a derivative, it often appears complex or unfriendly; this is where simplification comes into play. For simplification:

• Expand any parentheses or perform arithmetic operations in the numerator and denominator when necessary.
• Combine like terms to simplify the expression.
• Factor, if possible, to further reduce complex expressions.

In the exercise, after applying the quotient rule to find the first derivative \( f'(x) = \frac{(2x+3)(x-4) - (x^2+3x)(1)}{(x-4)^2} \), simplifying it to \( \frac{2x^2 - 5x - 12}{(x-4)^2} \) was necessary to make further calculus operations more manageable. The same process applies to the second derivative, ensuring that we have a clear and concise result. Simplifying derivatives ensures clarity in interpretation and is a critical step in solving calculus problems efficiently.

Calculus

Calculus is the branch of mathematics that studies continuous change and includes topics such as differentiation and integration. It's an essential tool for describing physical phenomena, optimizing systems, and predicting behaviors in various fields such as physics, engineering, and economics.

• Calculus allows us to calculate the rate of change through derivatives.
• It helps analyze functions by finding critical points, slope, and concavity.
• Provides techniques for finding the exact area under curves using integration.

In this exercise, calculus techniques like the quotient rule were employed to find both the first and second derivatives of the function \( f(x) = \frac{x^2 + 3x}{x-4} \). Through differentiation, calculus reveals detailed information about the behavior and properties of functions, showing how mathematics can describe and predict the world around us. This capacity for understanding dynamic systems and changes at an infinitesimal level often paints calculus as a cornerstone of modern mathematical applications and scientific inquiry.