Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=(3-5 x)^{5} $$

Step-by-Step Solution

Answer

\( \frac{d^3y}{dx^3} = -7500(3-5x)^2 \)

Step 1: Identify Inner Function

The function given is \( y = (3 - 5x)^5 \), which is a composition of functions. Here, the inner function is \( u = 3 - 5x \). This inner function will be used later to find derivatives using the chain rule.

Step 2: Differentiate the Outer Function

Differentiate \( y = u^5 \) with respect to \( u \). The derivative is \( \frac{dy}{du} = 5u^4 \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = 3 - 5x \) with respect to \( x \) to obtain \( \frac{du}{dx} = -5 \).

Step 4: Apply Chain Rule for First Derivative

To find the first derivative, \( \frac{dy}{dx} \), use the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 5u^4 \times (-5) = -25(3-5x)^4 \).

Step 5: Find the Second Derivative

Differentiate \( \frac{dy}{dx} = -25(3-5x)^4 \) with respect to \( x \) again. Use the chain rule: \( \frac{d^2y}{dx^2} = -25 \times 4(3-5x)^3 \times (-5) = 500(3-5x)^3 \).

Step 6: Find the Third Derivative

Now differentiate \( \frac{d^2y}{dx^2} = 500(3-5x)^3 \) once more with respect to \( x \). Apply the chain rule: \( \frac{d^3y}{dx^3} = 500 \times 3(3-5x)^2 \times (-5) = -7500(3-5x)^2 \).

Key Concepts

chain rule

The chain rule is a powerful tool in calculus for finding the derivative of composite functions. When you have a function nested inside another function, the chain rule becomes your go-to strategy. To apply it, you first identify the inner and outer functions.
- **Inner function**: This is the function inside, usually seen in the parentheses. In our exercise, it's \( u = 3 - 5x \).- **Outer function**: This is the entire function where the inner function is used as a variable. Here, it's \( y = u^5 \). To find the derivative using the chain rule, you first differentiate the outer function with respect to the inner function, then differentiate the inner function with respect to \( x \), and finally multiply these derivatives together. This systematic approach greatly simplifies differentiation of complex functions.

composition of functions

Understanding the composition of functions is crucial when differentiating nested functions. In essence, a composite function is formed when one function is applied to the result of another function.
In mathematics, this can be written as \( y = f(g(x)) \), where \( f \) is the outer function and \( g \) is the inner function. Our example demonstrates this with \( y = (3 - 5x)^5 \), where \( 3 - 5x \) is the inner function and the exponentiation is the outer function.
- **Role of composition**: Recognizing compositions helps you in applying the right rules like the chain rule.- **Effect on differentiation**: Each component of the composition needs to be handled individually and then pieced together using specified rules.Differentiating composites without this insight would be incredibly challenging.

differentiation steps

Differentiating functions systematically involves several clear steps. Let's break down the differentiation steps seen in the example.
1. **Identify Inner and Outer Functions**: Recognize which part of the function is inside and which is outside. For \( y = (3 - 5x)^5 \), \( 3 - 5x \) is the inner function and \( u^5 \) is the outer function.2. **Differentiate Separately**: Start by differentiating the outer function, \( \, \frac{dy}{du} = 5u^4 \, \), and the inner function \( \, \frac{du}{dx} = -5 \, \).3. **Apply the Chain Rule**: Multiply their derivatives for the first derivative: \( \, \frac{dy}{dx} = 5u^4 \times (-5) = -25(3-5x)^4 \, \).4. **Successive Derivatives**: Repeat the process: - For the second derivative: Differentiate \( \, -25(3-5x)^4 \, \) to get \( \, 500(3-5x)^3 \, \). - For the third derivative: Differentiate \( \, 500(3-5x)^3 \, \) finally yielding \( \, -7500(3-5x)^2 \, \).By following these steps, differentiating even the most complex expressions becomes manageable.