Question

Compute the derivative of the given function. $$f(x)=(2-3 x)^{2}$$

Short answer

The derivative of \(f(x) = (2 - 3x)^2\) is \(18x - 12\).

Step-by-step solution

Identify the Appropriate Rule

The function given is \(f(x)=(2-3x)^2\). This function is suitable for the chain rule, as it involves a composition of functions: an "outer" function \(u^2\) and an "inner" function \(u = 2 - 3x\).

Derive the Outer Function

The outer function is \(v(u) = u^2\). The derivative with respect to \(u\) is \( \frac{dv}{du} = 2u\).

Derive the Inner Function

The inner function is \(u(x) = 2 - 3x\). The derivative with respect to \(x\) is \( \frac{du}{dx} = -3\).

Apply the Chain Rule

According to the chain rule, the derivative of \(f(x) = v(u(x))\) is \( \frac{dv}{du} \cdot \frac{du}{dx} \). Substituting the expressions from the previous steps, we have \( \frac{d}{dx}(2-3x)^2 = 2(2-3x)(-3) \).

Simplify the Expression

Multiply the constants and the terms: \( \frac{d}{dx}(2-3x)^2 = 2(-3)(2-3x) = -6(2-3x) \). Simplify further to get \( -6 \cdot 2 + 6 \cdot 3x = -12 + 18x \).

Final Expression

The simplified expression for the derivative is \(f'(x) = 18x - 12\).

Key concepts

Derivative

A derivative represents the rate at which a function is changing at any given point. When we talk about the derivative \(f'(x)\), we're examining how a small change in \(x\) leads to a change in the function \(f(x)\).
The derivative is often thought of as the "slope" of the function's graph at a particular point. For example, in the provided exercise where \(f(x)=(2-3x)^2\), we are finding how the function's value changes when \(x\) changes slightly.
In our example, we first find derivatives of smaller components of the composite function, using the chain rule. This allows us to piece together these small changes and understand the overall rate of change of the function.

Composition of Functions

Composition of functions is the process of applying one function to the results of another function. It's like a chain of processes where the output of one becomes the input of another. This is common in mathematics where complex expressions are broken down into simpler functions.
In the problem with \(f(x)=(2-3x)^2\), we see two connected functions at work: an outer function, \(u^2\), and an inner function, \(u = 2 - 3x\).
The chain rule is pivotal here because it provides a seamless way to differentiate these compositions. Essentially, you're working from the outermost function progressing inward to the simplest components. This layered approach simplifies the process of finding the derivative for such intricate functions by using the derivatives of these smaller functions.

Simplification of Expressions

Simplification is the art of reducing an expression to its most straightforward form. It means breaking down and combining terms to present a cleaner, more understandable expression.
After applying the chain rule in our derivative example, the expression \(-6(2-3x)\)\ becomes the resulting product of differentiating the components. This still needs a final touch.
By distributing and combining like terms, we move to \(-6 imes 2 + 6 imes 3x\), which simplifies to \(-12 + 18x\).

• Breaking down this step helps prevent errors and confusion.
• Clear and concise final expressions make the derivative readily usable in further calculations.

This simplification is the final polish to get the neatest form of the derivative, \ f'(x)=18x-12\, which perfectly delineates how the function \(f(x)\) behaves around any value of \(x\).