Question

Find \(D_{x} y\) using the rules of this section. $$ y=(-3 x+2)^{2} $$

Short answer

The derivative is \( D_x y = 18x - 12 \).

Step-by-step solution

Identify the General Formula

The function given is in the form of \( y = (u(x))^n \) where \( u(x) = -3x + 2 \) and \( n = 2 \). We will use the chain rule to differentiate this, which states \( D_x (u(x))^n = n(u(x))^{n-1} \cdot u'(x) \).

Differentiate the Inner Function

First, find the derivative of the inner function \( u(x) \). For \( u(x) = -3x + 2 \), the derivative, using basic differentiation rules, is \( u'(x) = -3 \) since the derivative of \(-3x\) is \(-3\) and the derivative of a constant is zero.

Apply the Chain Rule

Now apply the chain rule already established. Substitute \(u(x)\) and \(u'(x)\) along with \(n\) into the formula: \[ D_x (u(x))^n = 2(u(x))^{2-1} \cdot u'(x) \]. Therefore, \[ D_x (-3x + 2)^2 = 2(-3x + 2) \cdot (-3) \].

Simplify the Expression

Simplify the expression: \[D_x (-3x + 2)^2 = 2(-3) (-3x + 2) = -6(-3x + 2). \]Expand the expression: \[-6(-3x + 2) = 18x - 12. \]

Write the Final Answer

The derivative of the function \(y=(-3x+2)^{2} \) is \[ D_x y = 18x - 12 \].

Key concepts

Differentiation

Differentiation is a core concept in calculus that deals with finding how a function's output changes as its input changes. This involves calculating the derivative of a function, which provides the rate of change or the slope of the function at any given point.
Understanding differentiation is crucial because it allows us to analyze and predict the behavior of variables across various fields like physics, engineering, and economics.
Differentiating a function involves applying various rules, like the power rule, product rule, and chain rule, depending on the form of the function one is dealing with.

Derivative

In calculus, the derivative of a function is a fundamental tool that measures how a function changes as its input varies. It is expressed as \(D_x y\), where \(y\) is the original function in terms of \(x\).
The derivative gives useful information like the slope of a curve at any point, indicating whether the function is increasing, decreasing, or remaining constant.
For algebraic functions such as \(y=(-3x+2)^{2}\), applying derivatives involves rules like the chain rule, which simplifies the process of finding how the output changes when the input is modified.

Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided into differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, learning how functions change, and determining slopes of curves.
The process of differentiation, especially using the chain rule, is a key aspect of differential calculus. This is incredibly useful for complex functions, where you can simplify the process by breaking the function into composite functions.
With calculus, we can model dynamic systems, optimize outputs, and solve real-world problems involving changing systems.

Function

A function in mathematics is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to precisely one output.
In the given exercise, \(y=(-3x+2)^2\) is a function where \(y\) depends on the variable \(x\).
Understanding functions is key to solving calculus problems, as it allows us to understand how variables interact within a problem. Functions can be modified using operations like addition, multiplication, and differentiation, revealing insights into the behavior of the relationship between the variables involved.