Question

Differentiate the following seven functions, applying the process of \(\$ 1\) step by step. \(y=4 x^{3}\)

Short answer

Answer: The derivative of the function \(y = 4x^3\) with respect to x is \(y' = 12x^2\).

Step-by-step solution

Identify the function's components

The given function is \(y = 4x^3\). Here, the coefficient (a) is 4, the base of the power (x) is x, and the exponent (n) is 3.

Apply the power rule

Using the power rule for differentiation, we find the derivative of the function: \(y' = nax^{n-1}\) For our function, a = 4, n = 3, and x is the variable. We plug these values into the power rule: \(y' = 3(4)x^{3-1}\)

Simplify the derivative

Now, we simplify the expression for the derivative: \(y' = 12x^{2}\) So, the derivative of the given function \(y = 4x^3\) is \(y' = 12x^{2}\).

Key concepts

Power Rule

The power rule is a fundamental principle in calculus and is used extensively for differentiation. It provides a straightforward method of finding the derivative of functions that are polynomials. The power rule states that if you have a function of the form \(y = ax^n\), where \(a\) and \(n\) are constants, then the derivative, \(y'\), is given by multiplying the exponent \(n\) by the coefficient \(a\), and then decreasing the exponent by one.

This can be expressed mathematically as:

• \(y' = nax^{n-1}\)

Consider the exercise function \(y = 4x^3\), where \(a = 4\) and \(n = 3\). By applying the power rule, we multiply 3 by 4 to get 12, and reduce the exponent from 3 to 2. This simplifies to \(y' = 12x^2\).

The power rule simplifies the process of taking derivatives, especially when dealing with higher-order polynomials.

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. Essentially, the derivative of a function is a measure of how the function value changes as its input changes. It is a fundamental concept, as it helps in understanding the behavior of functions, particularly the slope of their graphs.

For the function \(y = 4x^3\), the derivative \(y'\) is \(12x^2\). This means that at any point on the curve of \(y = 4x^3\), the rate of change, or slope, is determined by \(12x^2\).

This is why understanding derivatives is so important:

• They help in finding the rate of change.
• They provide instant rates, or instantaneous velocity, in physical problems.
• They serve as a tool for optimization, identifying maximum and minimum points in various applications.

In essence, derivatives enable a deeper comprehension of both the behavior and dynamics of mathematical functions.

Calculus

Calculus is a branch of mathematics dealing with continuous change. It's comprised of two main concepts: differentiation and integration. Differentiation, which was highlighted in the original exercise, is all about finding derivatives that deal with instantaneous change, as expressed in a curve's slope or a surface's gradient.

One of the reasons why calculus is so powerful is because it provides tools to model and solve problems involving change. Whether it’s physics, economics, engineering, or beyond, calculus assists in analyzing systems that undergo fluctuation.

Differentiation offers insight into:

• The behavior of function graphs through their slopes and curvature.
• Understanding systems in motion, such as an object’s acceleration.
• Solving real-world problems where knowing instant rates is crucial.

The simplicity of the power rule, applied in finding derivatives, exemplifies the practicality of calculus in solving complex mathematical functions with ease.