Problem 223 - Calculus Volume 1

Question

For each of the following exercises, a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and b. find \(\frac{d y}{d x}\) as a function of \(x .\) $$y=\left(\frac{x}{7}+\frac{7}{x}\right)^{7}$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 7\left(\frac{x}{7} + \frac{7}{x}\right)^6 \left(\frac{1}{7} - \frac{7}{x^2}\right) \)

Step 1: Identify Inner and Outer Functions

The given function is \( y = \left(\frac{x}{7} + \frac{7}{x}\right)^{7} \). We decompose it into two functions: the outer function \( f(u) = u^7 \) and the inner function \( u = g(x) = \frac{x}{7} + \frac{7}{x} \).

Step 2: Differentiate the Outer Function with respect to Inner Function

Differentiate \( f(u) = u^7 \) with respect to \( u \). Using the power rule, we get \( \frac{d}{du}f(u) = 7u^6 \).

Step 3: Differentiate the Inner Function with respect to x

Differentiate \( u = g(x) = \frac{x}{7} + \frac{7}{x} \) with respect to \( x \): \[ \frac{d}{dx} \left( \frac{x}{7} + \frac{7}{x} \right) = \frac{1}{7} - \frac{7}{x^2}. \]

Step 4: Apply Chain Rule to Find \( \frac{dy}{dx} \)

According to the chain rule, \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. \] Substituting in our derivatives: \( \frac{dy}{dx} = 7u^6 \cdot \left( \frac{1}{7} - \frac{7}{x^2} \right). \)

Step 5: Substitute Back the Expression for u

Substitute back \( u = \frac{x}{7} + \frac{7}{x} \) to get the derivative in terms of \( x \):\[ \frac{dy}{dx} = 7\left(\frac{x}{7} + \frac{7}{x}\right)^6 \cdot \left( \frac{1}{7} - \frac{7}{x^2} \right). \]

Key Concepts

Function Decomposition

When confronted with a complex function like \( y = \left(\frac{x}{7} + \frac{7}{x}\right)^{7} \), one useful technique is function decomposition. This involves breaking down the complicated expression into simpler functions that are easier to handle. We aim to express the original function in the form \( y = f(u) \) where \( u = g(x) \).
This is done by identifying an outer function and an inner function. Here, the outer function is \( f(u) = u^7 \), where \( u \) is a placeholder for the inner function \( g(x) = \frac{x}{7} + \frac{7}{x} \). Function decomposition simplifies differentiation, as each part can be treated separately before combining them using the chain rule.

Power Rule

The power rule is an essential tool in differentiation that allows us to find derivatives of functions raised to a power. It states that for any function \( u^n \), where \( n \) is a constant, the derivative with respect to the variable base is \( n \cdot u^{n-1} \).
In our exercise, the function \( f(u) = u^7 \) is differentiated using the power rule:

• The variable \( u \) is raised to the power of 7.
• The derivative of \( u^7 \) is \( 7u^6 \) by applying the rule.

The power rule simplifies the process of finding derivatives for polynomial expressions, making the chain of operations in calculus more manageable.

Differentiation

Differentiation is the process of determining the rate at which a function changes. When you differentiate a function, you calculate its derivative. This tells us how the function's output value changes in response to changes in the input value.
In this exercise, we perform differentiation on both the outer function \( f(u) = u^7 \) and the inner function \( u = \frac{x}{7} + \frac{7}{x} \). Steps include:

• Differentiate \( f(u) \) with respect to \( u \): \( \frac{d}{du}f(u) = 7u^6 \).
• Differentiate \( u = \frac{x}{7} + \frac{7}{x} \) with respect to \( x \): \( \frac{d}{dx}g(x) = \frac{1}{7} - \frac{7}{x^2} \).

Combining these results using the chain rule gives the complete derivative in terms of \( x \).

Inner and Outer Functions

In the context of differentiation, identifying inner and outer functions is crucial for applying the chain rule effectively. The outer function is the main framework of the equation, like raising a base to a power or applying a specific operation. The inner function modifies the variable before it's fed into the outer function.
For the expression \( y = \left(\frac{x}{7} + \frac{7}{x}\right)^{7} \):

• The outer function is \( f(u) = u^7 \), working on the result of the inner function.
• The inner function is \( u = g(x) = \frac{x}{7} + \frac{7}{x} \), that substitutes into the outer function.

By decomposing functions into inner and outer components, you streamline the differentiation process, enabling the use of the chain rule to determine derivatives accurately.