Problem 14 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=x^{3}+3 x^{2} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 3x^{2} + 6x \).

Step 1: Identify the Function Components

Observe the function given: \( y = x^{3} + 3x^{2} \). This function is a polynomial. We will use the rules of differentiation to find its derivative.

Step 2: Differentiate Each Term Separately

To differentiate the function, apply the power rule, which states \( \frac{d}{dx}(x^n) = nx^{n-1} \). Differentiating \( x^3 \) gives \( 3x^{2} \) and differentiating \( 3x^2 \) gives \( 6x \).

Step 3: Write the Derivative

Combine the differentiated terms from Step 2. The derivative is \( \frac{dy}{dx} = 3x^{2} + 6x \).

Step 4: Final Step: Review the Result

Ensure each step follows from the differentiation rules. The derivative of the function \( y = x^{3} + 3x^{2} \) is \( \frac{dy}{dx} = 3x^{2} + 6x \).

Key Concepts

Polynomial Functions

Polynomial functions represent a critical area of algebra and calculus. They are expressions consisting of variables and coefficients, combined using only operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are expressed in the form:

• \( y = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

In this form, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(x\) represents the variable. The highest power of the variable within a polynomial is called its degree. In the given exercise, the polynomial \( y = x^3 + 3x^2 \) is of degree 3.
Understanding polynomial functions forms the basis for many calculus concepts, especially when dealing with differentiation, as polynomials are smooth and continuous, making them relatively simple to differentiate.

Power Rule

The power rule is an essential principle in differentiation, particularly for polynomial functions. It allows for the swift calculation of derivatives of functions of the form \(x^n\). This rule states:

• \( \frac{d}{dx}(x^n) = nx^{n-1} \)

The exercise we reviewed uses this powerful rule to find derivatives. By applying the power rule, each term of a polynomial can be analyzed separately, making the differentiation process straightforward.
For the term \( x^3 \), applying the power rule gives \( 3x^2 \), while for \( 3x^2 \), it results in \( 6x \). It simplifies the process significantly and is a fundamental tool for anyone studying calculus.

Calculus

Calculus is a broad field of mathematics focusing on rates of change and accumulation of quantities. It is divided mainly into differential calculus and integral calculus. This exercise deals with differential calculus, specifically.

• Differentiation is the process used to find the derivative of a function. It essentially measures how a function changes as its input changes.

This foundational aspect of calculus is used across various fields such as physics, engineering, and economics to model and solve real-world problems.
Understanding calculus basics, like differentiation and the application of the power rule, is essential. This knowledge allows you to tackle complex problems with confidence and is crucial for advanced mathematical study.