Problem 10 - 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. $$ h(x)=\frac{x^{2}}{x+3} \quad\left(-1, \frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The value of the derivative of \(h(x)\) at the point (-1, 1/2) is -1. The differentiation rule used to find the derivative is the Quotient Rule.

Step 1: Applying the Quotient Rule

The quotient rule states that the derivative of a quotient of two functions is \((g(x)f'(x) - f(x)g'(x)) / (g(x))^2\), where f(x) is the numerator function and g(x) the denominator. Applying this to function \(h(x)\), let's set \(f(x) = x^2\) and \(g(x) = x+3\). Then, find their derivatives: \(f'(x) = 2x\) and \(g'(x) = 1\). Using the quotient rule: \(h'(x) = \((x+3)(2x) - (x^2)(1)\) / \((x+3)^2\)\).

Step 2: Simplify the Derivative

Simplify the resulting equation: \(h'(x) = \((2x^2+6x-x^2)\) / \((x+3)^2\) = \(x^2+6x\) / \((x+3)^2\)\).

Step 3: Plug in the Given Point

The given point coordinates are (-1, 1/2). The x-value is the input value for the derivative function. So, plug \(x=-1\) into the derivative function: \(h'(-1) = ((-1)^2 + 6(-1)) / ((-1+3)^2) = -1\).

Key Concepts

Quotient Rule

The Quotient Rule is a handy tool in calculus for finding the derivative of a function that is the ratio of two differentiable functions. This rule is used when a function can be expressed as

• numerator: \( f(x) \)
• denominator: \( g(x) \)

The formula to find the derivative of a function \( h(x) = \frac{f(x)}{g(x)} \) is \(\frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \).
Here,

• \( f'(x) \) is the derivative of the numerator \( f(x) \)
• \( g'(x) \) is the derivative of the denominator \( g(x) \)

This rule allows us to differentiate more complex expressions that involve division easily. For the function \( \frac{x^2}{x+3} \), using the quotient rule simplifies the process of differentiation by clearly establishing how each component is affected.

Differentiation

Differentiation is a fundamental concept in calculus focused on finding the rate at which a function is changing at any given point. It's the process of computing a derivative and is essential in physics, engineering, and mathematics for analyzing changes.
When differentiating a function, we apply rules and techniques to find the slope of the tangent line at any point \( x \) on the function, which tells us the function's instantaneous rate of change.
In the given exercise, to differentiate the function \( h(x) = \frac{x^2}{x+3} \), we recognize it as a quotient, so we must apply the Quotient Rule. This involves taking derivatives of both the numerator and the denominator separately

• Derivatives for \( f(x) = x^2 \) is \( f'(x) = 2x \)
• Derivative for \( g(x) = x+3 \) is \( g'(x) = 1 \)

These derivatives are then plugged into the quotient formula to obtain the derivative of the entire function.

Function Evaluation

Function evaluation refers to computing the output of a function given a specific input, often to understand the behavior of the function at that particular input.
In the context of derivatives, evaluating a function at a certain point helps us find the slope of the tangent line at that point, which is the derivative's value. This information is critical in understanding how the function behaves locally at that point.
For the function \( h(x) = \frac{x^2}{x+3} \), after finding its derivative \( h'(x) = \frac{x^2 + 6x}{(x+3)^2} \), we evaluate it at \( x = -1 \).

• Substitute \( x = -1 \) into the derivative: \( h'(-1) = \frac{(-1)^2 + 6(-1)}{(-1+3)^2} = -1 \).

So, at \( x = -1 \), the slope of the tangent line to the function \( h(x) \) is \( -1 \), indicating a downward trend at that point.