Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=x^{2}+4 x^{3} $$

Short answer

The derivative of the function \(g(x) = x^{2} + 4x^{3}\) is \(g'(x) = 2x + 12x^{2}\).

Step-by-step solution

Identify the Terms

There are two terms within the function: \(x^{2}\) and \(4x^{3}\). Each term must be differentiated individually.

Differentiate Term 1

Applying the power rule to the first term \(x^{2}\), where \(n\) is 2, gives the derivative as \(2 * x^{2-1} = 2x\).

Differentiate Term 2

To differentiate the second term \(4x^{3}\), first take out the constant 4 and apply the power rule to \(x^{3}\). This gives a derivative of \(4 * 3 * x^{3-1} = 12x^{2}\).

Combine the Results

Combining the derivatives of both terms gives the overall derivative of the function \(g'(x) = 2x + 12x^{2}\).

Key concepts

Power Rule

The Power Rule is a fundamental technique in differentiation, especially useful when dealing with polynomial functions. It provides a quick way to find the derivative of terms in which the variable is raised to a power. The basic format of the Power Rule is: if you have a term in the form of \(x^n\), then its derivative is \(n \cdot x^{n-1}\).
For example, when differentiating \(x^{2}\), you identify \(n=2\). According to the Power Rule, the derivative is \(2 \cdot x^{2-1} = 2x\). This method simplifies the differentiation process significantly, making it a staple in calculus, especially for terms that are simple powers of \(x\).
In our original exercise, each term of the polynomial applies the Power Rule to find their individual derivatives. Once you understand the Power Rule, solving for derivatives becomes much faster and more intuitive.

Derivative

In calculus, the derivative measures how a function's output changes as its input changes. It represents the rate of change or the slope at any point on a function. This concept is central to understanding how functions behave and is a key concept in fields such as physics, engineering, and economics.
The derivative is often denoted as \( f'(x) \) or \( \frac{d}{dx} \) for a function \( f(x) \). For a single-variable function like \(g(x) = x^{2} + 4x^{3}\), its derivative \(g'(x)\) tells you how steep the graph is at any point \(x\).
Finding the derivative involves identifying terms in the function and applying rules like the Power Rule to each term individually. Combining these results gives the derivative of the entire function. Understanding derivative calculations is crucial for mastering more complex mathematical and practical problems.

Polynomial Functions

Polynomial functions are expressions composed of variables (often \(x\)) raised to whole-number powers and multiplied by coefficients. These functions take the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0\). They are characterized by smooth, continuous curves and are easier to work with when differentiating.
In our exercise, the function \(g(x) = x^{2} + 4x^{3}\) is a polynomial function, composed of two terms: \(x^{2}\) and \(4x^{3}\). Each of these terms is a monomial, which contributes to the simplicity of using the Power Rule.
The polynomial's degree is determined by its highest power term (in this case, \(4x^3\)), indicating that this is a third-degree polynomial. Identifying the degree of a polynomial is important as it tells us about the general shape of its graph and behavior at extreme values of \(x\). By knowing such properties, you can predict the function's behavior without extensive calculations.