Problem 27 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(h(x)=(6 x-x^{3})^{2}\) is \(h'(x) = 2(6x - x^{3}) \cdot (6 - 3x^{2})\).

Step 1: Identify the Inner and Outer Functions

In the function \(h(x)=(6 x-x^{3})^{2}\), the inner function is \(g(x) = 6x - x^{3}\) and the outer function is \(f(u) = u^{2}\). So it could be interpreted as \(h(x) = f(g(x))\).

Step 2: Applying the Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. So, \[h'(x) = f'(g(x)) \cdot g'(x)\]

Step 3: Differentiate the Outer Function

The derivative of the outer function \(f(u)=u^{2}\) is \(f'(u) = 2u\). Replace \(u\) with the inner function: \(f'(g(x)) = 2(6x - x^{3})\).

Step 4: Differentiate the Inner Function

The derivative of the inner function \(g(x) = 6x - x^{3}\) is \(g'(x) = 6 - 3x^{2}\).

Step 5: Combine the Results

Multiply the results from step 3 and step 4 to get the derivative of the original function: \(h'(x) = f'(g(x)) \cdot g'(x) = 2(6x - x^{3}) \cdot (6 - 3x^{2})\).

Key Concepts

General Power Rule

The General Power Rule is a fundamental concept in calculus that simplifies the process of taking derivatives of functions raised to a power. This rule is an extension of the basic power rule and is essential when dealing with polynomials and other functions in the form of f(x) = [g(x)]^n, where g(x) is a differentiable function and n is a real number.

Using the General Power Rule, we find the derivative of the function h(x) by first differentiating the outer function u^n (considering g(x) as u), which gives us nu^{n-1}. Then, we multiply this result by the derivative of the inner function g(x). This application not only streamlines calculations but also forms a part of the Chain Rule, which we use when functions are composed of other functions.

Composite Function

A composite function is created when one function is applied to the results of another function. In other words, if f(x) and g(x) are two functions, then the composite function f(g(x)) means that we apply g to x first and then apply f to the result of g(x). In our exercise example, h(x) = (6x - x^3)^2, the inner function g(x) is 6x - x^3, and the outer function is u^2, which is applied after g(x).

Understanding this relationship is crucial when working with derivatives since the Chain Rule directly applies to composite functions. Knowing how to identify the inner and outer functions is the first step in employing the Chain Rule effectively.

Derivative of a Function

The derivative of a function represents the rate of change or the slope of the function at any given point. It's one of the central concepts in differential calculus and is essential in various scientific and engineering fields. In essence, the derivative tells us how one variable changes concerning another.

For the function h(x) in our example, deriving both the outer and inner functions separately before applying the Chain Rule gives us a clear understanding of how h(x) changes with x. Practical application involves finding derivatives of simple functions first and then combining these derivatives following the rules of differentiation, such as the General Power Rule and the Chain Rule, to find the derivative of more complex functions or composite functions.