Problem 1 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=4 x^{7}$$

Step-by-Step Solution

Answer

The derivative is \(\frac{dy}{dx} = 28x^{6}\).

Step 1: Identify the form of the equation

The equation is given as \(y = 4x^7\). This is a basic power function, where the power rule can be applied to find the derivative.

Step 2: Apply the Power Rule

The power rule for differentiation states that if \(y = ax^n\), then \(\frac{dy}{dx} = anx^{n-1}\). Here, \(a = 4\) and \(n = 7\).

Step 3: Calculate the Derivative

Using the power rule, multiply the coefficient \(4\) by the exponent \(7\) to get the new coefficient. Then reduce the exponent by 1. This gives: \(\frac{dy}{dx} = 28x^{6}\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool in calculus used for finding derivatives of functions with power terms. When you have a function of the form \( y = ax^n \), the derivative is found by multiplying the exponent \( n \) with the coefficient \( a \), and then reducing the exponent by one. This process can be mathematically expressed as:

• If \( y = ax^n \), then \( \frac{dy}{dx} = anx^{n-1} \).

This rule is powerful because it simplifies the process of differentiation for polynomial expressions. For instance, in the exercise provided, the term \( 4x^7 \) can easily be differentiated by applying the power rule to obtain \( 28x^6 \). This demonstrates how the power rule streamlines complex calculations into a simple arithmetic process.

The Process of Differentiation

Differentiation is a core concept in calculus that involves computing the derivative of a function. The derivative represents how a function changes as its input changes - it is effectively the slope of the tangent to the curve of the function at any given point.

Using the power rule for differentiation simplifies the task of finding derivatives for polynomial functions. Begin by identifying the base form of the function. Confirm if the function fits the \( ax^n \) form. Then, carefully apply the power rule as described earlier.

• Multiply the exponent by the coefficient.
• Reduce the exponent by one.

This process ultimately reflects how the rate of change of the function behaves, making differentiation an essential tool in various fields, such as physics and engineering, where understanding change is crucial.

The Fundamental Role of Calculus

Calculus is the branch of mathematics concerned with continuous change. It is divided into differential and integral calculus. In this context, we focus on differentiation, a part of differential calculus, which deals with finding the rate of change.

This exercise on finding derivatives using the power rule is a classic example of calculus in action. Calculus equips us with techniques to solve problems involving motion, growth, and decay, which are constant phenomena in nature and technology.

• Derivatives help determine the optimized designs in engineering.
• They provide insights into rates of change in economics and natural sciences.

The mathematical foundation built through calculus is invaluable. It not only enhances problem-solving skills but also provides a deeper understanding of the behavior and interaction of dynamic systems.