Problem 7 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=-\frac{1}{3}\left(x^{7}+2 x-9\right)$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = -\frac{7}{3}x^6 - \frac{2}{3}\)

Step 1: Identify the components of the function y

The function is given as \( y = -\frac{1}{3}(x^7 + 2x - 9) \). We need to identify it as a whole and its individual parts: the terms within the parentheses \( x^7 \), \( 2x \), and the constant \(-9\).

Step 2: Apply the derivative to the function

To find \( \frac{dy}{dx} \), we apply the derivative operator to each term inside the parentheses. The differentiation operation is linear, so it distributes over the sum/difference inside the parentheses. Thus: \[ \frac{d}{dx} \left(-\frac{1}{3}(x^7 + 2x - 9)\right) = -\frac{1}{3} \left( \frac{d}{dx}(x^7) + \frac{d}{dx}(2x) + \frac{d}{dx}(-9) \right) \]

Step 3: Differentiate each term individually

Differentiate each of the terms separately: - The derivative of \(x^7\) is \(7x^6\) (using the power rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\)).- The derivative of \(2x\) is \(2\) (the derivative of a constant times \(x\) is just the constant).- The derivative of \(-9\) is \(0\) (constants have a derivative of zero).

Step 4: Combine the differentiated terms

Substitute the derivatives back into the expression from Step 2: \[ -\frac{1}{3} (7x^6 + 2 + 0) \] Simplify to get: \[ -\frac{1}{3} (7x^6 + 2) \]

Step 5: Expand and simplify the expression

Distribute \(-\frac{1}{3}\) through the terms inside the parentheses: \[ = -\frac{1}{3} \cdot 7x^6 - \frac{1}{3} \cdot 2 \] This simplifies to: \[ -\frac{7}{3}x^6 - \frac{2}{3} \]

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives. It simplifies the process of differentiation for power functions. Specifically, if you have a function of the form \( f(x) = x^n \), the power rule states that the derivative of this function is \( f'(x) = nx^{n-1} \). This means you multiply the exponent by the base raised to the power of one less than the original exponent.

• This rule is rapid and effective for functions that are simple monomials or sums of monomials.
• It's important to remember that the power rule applies directly only where \( n \) is a real number.

For example, in the problem given, the power rule was used to find the derivative of the term \( x^7 \), resulting in \( 7x^6 \). Each term is handled individually, making the process systematic and straightforward.

Differentiation

Differentiation is the process of finding the derivative of a function. It tells us how a function changes as its input changes, which is crucial in understanding the behavior of mathematical functions. The derivative itself is a function that outputs the rate of change of the original function at any given point.

• Differentiation is a linear operation, meaning it applies to each term separately and respects the sum or difference between terms.
• Constants and linear multiples (like the \(-\frac{1}{3}\) in the exercise) factor out during differentiation.

In the exercise, differentiation was applied separately to \( x^7 \), \( 2x \), and \(-9\). Constants differentiate to zero because they do not change, while terms like \( 2x \) simplify directly to \( 2 \). Understanding how to apply differentiation, especially in complex expressions, is essential in calculus.

Calculus

Calculus is the branch of mathematics focused on change. It encompasses differentiation and integration, which are tools for investigating and managing changing quantities. While differentiation deals with finding instantaneous rates of change, its counterpart, integration, is concerned with accumulations of quantities and area under curves.

• Calculus provides a framework for modeling physical systems where variables change with time or other influences.
• It is widely used in sciences, economics, engineering, and beyond for modeling and solving real-world problems.

In the context of the exercise, calculus not only provided the method (differentiation) to solve the function's rate of change but also showed us a way to systematically understand how each piece of the expression behaves. Mastery of calculus principles allows students and professionals to solve complex problems effectively.