Question

Compute the derivative of the given function. $$g(x)=(2 x-5)^{3}$$

Short answer

The derivative is \(6(2x-5)^2\).

Step-by-step solution

Identify the Outer and Inner Functions

The given function \[g(x)=(2x-5)^3\]is a composition of functions. Here, we can identify the outer function as \(u^3\) and the inner function as \(u = 2x - 5\). We will use the chain rule to find the derivative.

Differentiate the Outer Function

The derivative of the outer function \(u^3\) with respect to \(u\) is \[\frac{d}{du}(u^3) = 3u^2.\]This means that when applying the chain rule, we'll multiply this derivative by the derivative of the inner function.

Differentiate the Inner Function

Now, differentiate the inner function \(u = 2x - 5\) with respect to \(x\). This gives us:\[\frac{d}{dx}(2x - 5) = 2.\]

Apply the Chain Rule

According to the chain rule, the derivative of \(g(x) = (2x-5)^3\) is:\[\frac{dg}{dx} = \frac{d}{du}(u^3) \cdot \frac{du}{dx} = 3u^2 \cdot 2.\]Substitute back \(u = 2x - 5\) to get:\[\frac{dg}{dx} = 3(2x-5)^2 \cdot 2.\]

Simplify the Expression

Now simplify the expression:\[\frac{dg}{dx} = 6(2x - 5)^2.\]This is the derivative of the function \(g(x) = (2x - 5)^3\).

Key concepts

Derivative

In calculus, the derivative is a fundamental concept that measures how a function changes as its input changes. The derivative indicates the rate of change or the slope of the function at a particular point. It gives you the velocity of a function, so to speak, which can be thought of as the function's 'speed' with respect to changes in its input variable.
In the exercise, we needed to find the derivative of the function \(g(x) = (2x-5)^3\). Here, the derivative \(\frac{dg}{dx}\) represents how \(g(x)\) changes as \(x\) changes.
To compute this derivative, we apply certain differentiation rules, such as the power rule and the chain rule. Each rule helps us handle different structures of functions and makes differentiation simpler.

Chain Rule

The chain rule is an essential derivative rule in calculus that is used for finding the derivative of a composite function. A composite function is created when one function is nested inside another function.
The chain rule states that if you have a composite function \(f(g(x))\), the derivative \(\frac{d}{dx}[f(g(x))]\) is calculated by differentiating the outer function \(f\) with respect to the inner function \(g\), and then multiplying by the derivative of the inner function \(g\) with respect to \(x\).
This can be summarized as:

• Find the derivative of the outer function.
• Multiply it by the derivative of the inner function.

In our given problem, \(g(x) = (2x-5)^3\), the outer function is \(u^3\) and the inner function is \(u = 2x - 5\). Calculating the derivative involves differentiating both and applying the chain rule to link them together. This allows us to compute the derivative of the composite function efficiently.

Composition of Functions

Understanding the composition of functions is crucial when dealing with problems like finding derivatives of composite functions. The composition \(f(g(x))\) signifies applying a function \(f\) to the result of another function \(g(x)\).
In simpler terms, you can think of it like a mathematical 'function machine' where \(g(x)\) is processed first, and its output is then fed into \(f\).
This layered structure is what makes the chain rule necessary, as it directly addresses how derivatives behave within such compositions.
In the exercise, \(g(x) = (2x-5)^3\) is a composition of two functions: an inner linear function \(2x-5\) and an outer power function that raises its input to the third power. Acknowledging this structure allows us to correctly apply the chain rule, ensuring that each component is differentiated properly in relation to the others.