Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Demand In Exercises, find the rate of change of \(x\) with respect to \(p\). $$ p=\frac{2}{0.00001 x^{3}+0.1 x} \quad x \geq 0 $$

Step-by-Step Solution

Answer

The rate of change of \(x\) with respect to \(p\) is \(\frac{dp}{dx}= \frac{-2(0.00003x^2+0.1)}{((0.00001x^3+0.1x)^2)}\)

Step 1: Rewrite the formula for p in terms of x

We can rewrite the given formula for \(p\) as \(p = 2 \left( 0.00001x^{3} + 0.1x \right)^{-1}\) to make the differentiation process more straightforward.

Step 2: Differentiate p with respect to x using the chain rule

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Thus, \(\frac{dp}{dx} = -2(0.00001x^3+0.1x)^{-2} \times (0.00003x^2+0.1)\).

Step 3: Simplify the derivative formula

We can simplify this derivative formula to get \(\frac{dp}{dx}= \frac{-2(0.00003x^2+0.1)}{((0.00001x^3+0.1x)^2)}\)

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It is used to determine how a function's output value changes as its input changes. If a function describes a curve, differentiation tells us how steep the curve is at any given point. This concept is crucial in understanding real-world phenomena where change is involved, such as velocity, rates of supply and demand, or even cost variations.

In mathematical terms, differentiation involves calculus operations to find the derivative. The derivative symbol \(\frac{dy}{dx}\) represents this rate of change, where \(y\) is a function of \(x\). The process of differentiation finds this symbol in terms of other known variables, enabling us to predict changes.

When approaching differentiation, especially with complex functions, it is important to rewrite them into simpler forms. This simplification is one of the first steps, as seen in the exercise, where the function was restated in a more recognizable format. This makes applying differentiation rules like the chain rule more straightforward.

Chain Rule

The chain rule is an essential tool in differentiating composite functions. Composite functions are those where one function is inside another. For example, if you encounter \(g(x) = f(u(x))\), you can differentiate it using the chain rule.

The chain rule formula is: \(\frac{d}{dx}[f(u(x))] = f'(u(x)) \cdot u'(x)\). This means you differentiate the outer function while leaving the inner function unchanged and multiply by the derivative of the inner function.

• First, identify both the inner and outer functions.
• Differentiating the outer function involves treating the inner function as a simple variable.
• Next, differentiate the inner function.
• Finally, multiply these two derivatives to find the final result.

In the exercise, understanding the chain rule is vital in differentiating the complex demand function relating \(p\) and \(x\). By breaking it down into these two simpler steps, the composite function can be managed easily. It simplifies the problem of finding the rate of change, making the process much more efficient.

Derivative

A derivative, in the simplest terms, is a measure of how a function changes. You can think of it as the slope of a line tangent to a curve at any point along it. This concept is foundational to calculus, helping us understand and predict changes in quantity.

In the context of real-world applications, a derivative can indicate speed, the slope of a hill, or even economic elasticity. It helps to answer questions of how, and how much, things change. For instance, the derivative in the exercise connects the change in price \(p\) of a product to its demand \(x\), providing a powerful insight into economic behavior.

• The derivative is often represented as \(\frac{dy}{dx}\) or \(f'(x)\).
• It is calculated using rules such as the power rule, product rule, quotient rule, and chain rule.
• The result guides us in understanding the nature and direction of change.

Having a firm grasp on derivatives, especially when combined with specific rules like the chain rule, equips you to tackle a wide variety of mathematical problems. This makes it possible to deconstruct complex functions, such as the one encountered in the exercise, into manageable elements to solve the rate of change effectively.