Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(d^{3} y / d x^{3}\). $$ y=(3-5 x)^{5} $$

Step-by-Step Solution

Answer

The third derivative is \(-7500(3-5x)^2\).

Step 1: Identify the Function and its Derivative Order

We are given the function \( y = (3 - 5x)^5 \) and need to find the third derivative, \( \frac{d^3y}{dx^3} \). This requires calculating derivatives up to the third order.

Step 2: First Derivative using Chain Rule

To find the first derivative \( \frac{dy}{dx} \), apply the chain rule: \( y = u^5 \) with \( u = 3 - 5x \). Thus, \( \frac{dy}{du} = 5u^4 \) and \( \frac{du}{dx} = -5 \). Therefore, \( \frac{dy}{dx} = 5(3-5x)^4 (-5) = -25(3-5x)^4 \).

Step 3: Second Derivative

For the second derivative, \( \frac{d^2y}{dx^2} = \frac{d}{dx}(-25(3-5x)^4) \), use the chain rule again. Here, \( u = (3-5x)\), so \( du/dx = -5 \) and the outer derivative gives us \(-100(3-5x)^3 (-5) = 500(3-5x)^3 \).

Step 4: Third Derivative

For the third derivative, \( \frac{d^3y}{dx^3} = \frac{d}{dx}(500(3-5x)^3) \). Again applying the chain rule with \( du/dx = -5 \), we have \( 1500(3-5x)^2 (-5) = -7500(3-5x)^2 \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus, used to find derivatives of composite functions. When you have a function nested within another, the chain rule becomes necessary. For instance, if you have a function like \(y = (3-5x)^5\), it's not straightforward to derive directly without considering its inner and outer components.

The basic idea of the chain rule is to differentiate the outer function first while keeping the inner function the same, and then multiply by the derivative of the inner function. This two-step process lets you tackle complex derivatives methodically.

• First, identify the inner function (\(u = 3 - 5x\) in our example) and differentiate it (\(du/dx\)).
• Second, differentiate the outer function as if the inner function was just a variable (\(dy/du\)).
• Finally, multiply these derivatives to find \(dy/dx\).

In our exercise, the chain rule helped find the first derivative by recognizing \((3-5x)^5\) as a function of another function (3-5x), leading to \(-25(3-5x)^4\) as the derivative of \(y\).

Calculus Problems

Calculus problems often involve multiple skills, such as differentiation, integration, and understanding limits. When tasked with finding the third derivative of a function like \((3-5x)^5\), you should break the process into manageable steps.

Begin by understanding what is required: calculating successive derivatives up to the third order. Each subsequent derivative relies on the previous one. In this scenario, each step involves applying the chain rule multiple times to handle the changes in function complexity.

• Identify what type of problem you are facing (in this case, finding higher-order derivatives).
• Use appropriate techniques, such as the chain rule, for each step.
• Ensure all calculations are meticulous and accurate to prevent errors cascading through the steps.

By breaking down the problem into smaller parts, the problem's complexity is more manageable, allowing you to focus on simpler tasks at each step.

Derivative Calculation

Derivative calculation is the process of finding the rate at which a function changes. For complex expressions, this involves using rules like the chain rule repeatedly.

In our example, finding \(d^3y/dx^3\) required calculating the first, second, and finally the third derivatives. Each derivative brings us closer to understanding the function's behavior by considering its rate of change at progressively finer levels.

• The first derivative \(dy/dx = -25(3-5x)^4\) tells us the initial rate of change.
• The second derivative \(d^2y/dx^2 = 500(3-5x)^3\) provides insight into changes in the rate of change, often related to concavity.
• The third derivative \(d^3y/dx^3 = -7500(3-5x)^2\) indicates how the second derivative is changing over time.

Derivative calculations aren't just about finding numbers; they reveal the deeper dynamics of a function’s behavior, essential for comprehending complex calculus problems.