Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, find \(D_{x} y\). $$ y=\left(x^{2}-x+1\right)^{-7} $$

Step-by-Step Solution

Answer

The derivative is \(D_x y = -7(2x - 1)(x^2 - x + 1)^{-8}\).

Step 1: Identify the Derivative Rule

The function given is of the form \(y = (u(x))^{-n}\), where \(u(x) = x^2 - x + 1\) and \(n = 7\). To find \(D_x y\), we apply the chain rule. This involves differentiating \((x^2 - x + 1)^{-7}\) with respect to \(x\) using the chain rule and power rule.

Step 2: Differentiate the Outer Function

Using the power rule, the derivative of \((u(x))^{-n}\) is given by \(-n\cdot u(x)^{-n-1}\). Substituting the values, we get \(-7(u(x))^{-8}\). So, the derivative of the outer function is \(-7(x^2 - x + 1)^{-8}\).

Step 3: Differentiate the Inner Function

Now, differentiate \(u(x) = x^2 - x + 1\) with respect to \(x\). The derivative is \(2x - 1\).

Step 4: Apply the Chain Rule

According to the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. Therefore, \(D_x y = -7(x^2 - x + 1)^{-8} \cdot (2x - 1)\).

Step 5: Simplify the Expression

Simplify the derivative expression: \(D_x y = -7(2x - 1)(x^2 - x + 1)^{-8}\). This is the final simplified derivative.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of a composite function. In simple terms, a composite function is a function within another function. For example, if we have a function that looks like \(f(g(x))\), then to differentiate it, the chain rule comes into play.
The chain rule states that to find the derivative of \(f(g(x))\), you need to first take the derivative of the outer function \(f\) with respect to \(g\), and then multiply it by the derivative of the inner function \(g\) with respect to \(x\).

• This approach breaks down complex functions into manageable pieces.
• It's like peeling off layers of an onion—it allows us to handle each layer separately.

In the given exercise, the chain rule is employed to differentiate the function \(y = (x^2 - x + 1)^{-7}\). This is a perfect example of a composite function where \(x^2 - x + 1\) is inside the power, which makes it an ideal candidate for the chain rule.
Integrating the chain rule correctly will transform complex derivatives into simpler, step-by-step calculations.

Power Rule

The power rule is one of the simplest and most commonly used rules in differentiation, essential for handling polynomial functions. It states that the derivative of \(x^n\) is \(nx^{n-1}\). This rule applies to any real number exponent \(n\).

• It is particularly useful for powers of \(x\) that are positive, negative, fractional, or zero.
• The power rule simplifies derivatives involving powers efficiently.

In our exercise, the function \((u(x))^{-7}\) utilizes the power rule to handle its exponent part.
Here's how it is applied: when we have \(u(x)^{-n}\), the power rule tells us that the derivative will be \(-n\cdot u(x)^{-n-1}\). The power rule provides a straightforward way to deduce this result by just adjusting the exponent and multiplying by the base's derivative. Together with the chain rule, it forms the backbone of differentiating polynomials and exponential expressions.

Differentiation

Differentiation in calculus is the process used to find the rate at which a function is changing at any given point. In simpler terms, it provides us with information on how the y-value of a function changes with respect to changes in the x-value.

• Differentiation can tell us about the slope or gradient of a curve at a specific point.
• It's the foundation for many calculus operations and applications like finding tangents, velocities, and optimizations.

In the context of the exercise, differentiation is applied to both the outer function \((x^2-x+1)^{-7}\) and the inner function \(x^2-x+1\).
This dual differentiation process utilizes both the chain and power rules, demonstrating differentiation's versatility and importance in breaking down and solving complex expressions.

Mathematical Functions

Mathematical functions are expressions that relate an input to an output. In calculus, these functions often involve variables raised to a power, multiplied by constants, or combined through arithmetic operations.

• Functions provide a solid framework for modeling real-world problems.
• The type and form of a function determine the approach we use in differentiation.

The exercise focuses on a specific type of mathematical function, \(y = (x^2-x+1)^{-7}\), which is part of the broader family of polynomial and power functions.
Understanding the structure of mathematical functions is crucial because it guides us in selecting the appropriate differentiation technique, whether it’s the chain rule, power rule, or another method. Ultimately, identifying and working with different mathematical function types allows for flexible problem-solving strategies in calculus.