Problem 21 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ y=\left(4-x^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \(-8x(4-x^2)^3\).

Step 1: Identify the Outer and Inner Function

The function given is a composition of two functions: the outer function, which is \(u^4\), and the inner function, \(u = 4 - x^2\). Here, \(u\) is known as the inner function and \(y = u^4\) is the outer function.

Step 2: Differentiate the Outer Function

Differentiate the outer function \(y = u^4\) with respect to \(u\). This gives \(\frac{dy}{du} = 4u^3\).

Step 3: Differentiate the Inner Function

Differentiate the inner function \(u = 4 - x^2\) with respect to \(x\). This results in \(\frac{du}{dx} = -2x\).

Step 4: Apply the Generalized Power Rule

The Generalized Power Rule states that if \(y = (g(x))^n\), then \(\frac{dy}{dx} = n(g(x))^{n-1} \cdot g'(x)\). Thus, substituting what we have: \[\frac{dy}{dx} = 4(4-x^2)^3 \cdot (-2x)\]

Step 5: Simplify the Expression

Combine the terms to find the simplified form of the derivative: \[\frac{dy}{dx} = -8x(4 - x^2)^3.\]

Key Concepts

Generalized Power Rule

The Generalized Power Rule is an extension of the power rule, which allows us to take the derivative of a function raised to a power. It is especially useful when dealing with composite functions. In the Generalized Power Rule, we consider a function of the form \[ y = (g(x))^n \] Here, the rule states that the derivative of such a function is given by: \[ \frac{dy}{dx} = n(g(x))^{n-1} \cdot g'(x) \] This means we need to differentiate the outside part first, treating the inner function as a single entity, before multiplying by the derivative of the inner function. Common scenarios where this rule is applied include situations involving polynomial functions, trigonometric functions raised to powers, and any complicated nest of functions. The key is to break down the function into simpler parts that can be handled more easily.

Derivative

A derivative represents the rate of change of a function with respect to a variable. In simpler terms, it tells us how a function is changing at any given point, providing insight into the function's behavior. The derivative of a function is symbolized by different notations like \( \frac{dy}{dx} \), \( f'(x) \), or \( Df \).

• It measures how much the dependent variable changes when the independent variable changes slightly.
• Geometrically, it corresponds to the slope of the tangent line at a particular point on the graph of the function.
• It's fundamental in calculus for finding maximum and minimum values of functions, creating linear approximations, and solving problems involving motion and change.

To find the derivative, we apply various rules of differentiation like the Power Rule, Product Rule, Quotient Rule, and others depending on the type of function.

Composition of Functions

Composition of functions involves plugging one function into another. If you have functions \( f(x) \) and \( g(x) \), the composition \( f(g(x)) \) means you are substituting \( g(x) \) into \( f(x) \). This concept is crucial in calculus because it often appears in complex problems.

• The function inside (\( g(x) \)) is known as the inner function.
• The function that wraps around the inner function (\( f(x) \)) is the outer function.

When differentiating, composed functions require the use of the chain rule, allowing us to manage the layers of functions effectively. Real-life examples of composition include calculating interest over time (where interest itself can be a function of time and other factors) or finding the temperature as a function of pressure and time.

Chain Rule

The Chain Rule is a method for differentiating compositions of functions. It's an essential tool in calculus, enabling the calculation of derivatives for functions that can be expressed as a composition of two or more functions. When applying the chain rule, if you have a function \( y = f(g(x)) \), the derivative is given by: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] Steps to apply the Chain Rule include:

• Identify the outer function \( f \) and its derivative \( f' \).
• Determine the inner function \( g \) and its derivative \( g' \).
• Multiply the derivative of the outer function by the derivative of the inner function.

Using this rule helps in breaking complex problems into workable parts, making it easier to handle non-linear situations in physics, engineering, or any field that involves changes with respect to time or other variables.