Problem 2 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=-3 x^{12}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = -36x^{11} \).

Step 1: Identify the Power Rule

To find the derivative \( \frac{dy}{dx} \) of the function \( y = -3x^{12} \), you will use the power rule. The power rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \).

Step 2: Apply the Power Rule

Identify \( a = -3 \) and \( n = 12 \) in the function. Using the power rule, multiply the exponent by the coefficient: \( 12 \times (-3) = -36 \). Then, decrease the exponent by one, going from \( 12 \) to \( 11 \).

Step 3: Write the Derivative

Combine the results from the previous step to write the derivative of \( y = -3x^{12} \) as \( \frac{dy}{dx} = -36x^{11} \).

Key Concepts

Derivative

In mathematics, a derivative represents the rate at which a function is changing at any given point. This is a fundamental concept in calculus, essential for understanding how functions behave. The derivative of a function is often denoted as \( \frac{dy}{dx} \), which reads as "the derivative of \( y \) with respect to \( x \)." This tells us how \( y \) changes as \( x \) changes.

Understanding derivatives is essential as it provides insights into various real-world applications where change is involved. Some real-world applications include:

• Physics, where derivatives describe motion and rates of change, such as velocity and acceleration.
• Economics, where derivatives can indicate changes in cost, profit, or revenue.
• Biology, where derivatives can model population growth rates or decay in radioactive substances.

When we talk about deriving polynomial functions, like in our exercise, the Power Rule becomes particularly handy. Here, it simplifies finding derivatives efficiently.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to various powers, each multiplied by a constant coefficient. For example, in the expression \( y = -3x^{12} \), \( -3x^{12} \) is a single term polynomial, also known as a monomial.

Polynomials are:

• Made up of terms with a variable raised to a non-negative integer exponent.
• Characterized by their degree, which is the highest power of the variable in the expression.
• Commonly used in mathematics for modeling various types of data and forming equations.

Recognizing polynomial functions is crucial for applying calculus techniques effectively. These functions are smooth and continuous, making them easy to work with when finding derivatives using calculus rules like the Power Rule.

Calculus Techniques

Calculus is a branch of mathematics focused on change. To handle this, it uses various techniques. One popular and powerful technique in calculus is the Power Rule. This rule is specifically useful for differentiating polynomials quickly and efficiently.

To apply the Power Rule to a polynomial function \( y = ax^n \), you can follow these steps:

• Multiply the exponent \( n \) by the coefficient \( a \). This becomes the new coefficient.
• Reduce the exponent by one, resulting in \( n-1 \) for the power of \( x \).
• Combine these results to form the derivative.

For instance, when finding the derivative of \( y = -3x^{12} \), as shown in the exercise, you first multiply \( 12 \) (the exponent) with \(-3\) (the coefficient) to get \(-36\). Then, decrease the exponent from \( 12 \) to \( 11 \), resulting in \( \frac{dy}{dx} = -36x^{11} \).

Utilizing these calculus techniques not only simplifies computation but also deepens your understanding of how mathematical functions behave.