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Chapter 3: Problem 11

In Exercises \(9-18,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x .\) $$y=\left(1-\frac{x}{7}\right)^{-7}$$

Short Answer

Expert verified
\( \frac{dy}{dx} = \left(1 - \frac{x}{7}\right)^{-8} \)

Step by step solution

01

Identify the components of the function

Start by examining the given function, which is \(y = \left(1 - \frac{x}{7}\right)^{-7}\). The goal is to express this function in terms of a new function \(f(u)\) and an inner function \(g(x)\).
02

Define the inner function

Choose the expression inside the parentheses as the inner function. Set \(u = g(x) = 1 - \frac{x}{7}\). This simplifies the original function into an outer function based on \(u\).
03

Define the outer function

Express the original function in terms of \(u\). With \(u\) defined, \(y\) becomes a function of \(u\): \(y = f(u) = u^{-7}\).
04

Differentiate the outer function

Differentiate \(f(u) = u^{-7}\) with respect to \(u\) to find \(\frac{dy}{du}\). Use the power rule: \(\frac{dy}{du} = -7u^{-8}\).
05

Differentiate the inner function

Differentiate the inner function \(u = 1 - \frac{x}{7}\) with respect to \(x\) to find \(\frac{du}{dx}\). The derivative is \(\frac{du}{dx} = -\frac{1}{7}\).
06

Apply the Chain Rule

Use the chain rule to find \(\frac{dy}{dx}\). This is done by multiplying \(\frac{dy}{du}\) with \(\frac{du}{dx}\): \(\frac{dy}{dx} =\frac{dy}{du} \cdot \frac{du}{dx} = -7u^{-8} \cdot -\frac{1}{7}\).
07

Simplify the expression

Simplify the expression for \(\frac{dy}{dx}\). The \(-7\) and \(-\frac{1}{7}\) cancel out, yielding \(u^{-8}\). Substitute back \(u = 1 - \frac{x}{7}\), leading to \(\frac{dy}{dx} = \left(1 - \frac{x}{7}\right)^{-8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Composite Functions
In calculus, a composite function is a function that results from the combination of two other functions. These other functions are typically known as the "inner" and "outer" functions. When you have an equation like \( y = (1 - \frac{x}{7})^{-7} \), breaking it into a composite function can help simplify the differentiation process.
First, identify the inner function; here, it's \( g(x) = 1 - \frac{x}{7} \). This represents the expression inside the parentheses. The outer function, now relying on \( u \), becomes \( y = f(u) = u^{-7} \). This step is crucial as it separates the complexities of dealing with nested functions.
In practice:
  • Identify the inner and outer functions.
  • Express the original function as a function of the inner function.
Understanding composite functions and being able to decompose them into these parts is essential, especially when further differentiation is required.
Differentiation
Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. For a function \( f(x) \), the derivative \( f'(x) \) gives this rate of change. Calculating the derivative of a composite function uses the Chain Rule, a powerful tool in calculus.
In this exercise, after expressing \( y = (1 - \frac{x}{7})^{-7} \) as a composite function, you can differentiate both the inner and the outer functions separately. First, find \( \frac{du}{dx} \) for the inner function \( g(x) = 1 - \frac{x}{7} \), leading to \( \frac{du}{dx} = -\frac{1}{7} \). Then, differentiate the outer function \( f(u) = u^{-7} \) to get \( \frac{dy}{du} = -7u^{-8} \).
Differentiation steps:
  • Identify changes of the outer function concerning the inner variable.
  • Identify changes of the inner function concerning \( x \).
This ensures you鈥檙e set to apply the Chain Rule effectively.
Power Rule
The Power Rule is one of the simplest and most useful rules in differentiation, particularly when dealing with functions of the form \( x^n \). This rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
When differentiating the outer function in this problem, \( f(u) = u^{-7} \), the Power Rule simplifies the process. By applying it, you derive \( \frac{dy}{du} = -7u^{-8} \), which represents the derivative of \( u^{-7} \) with respect to \( u \). This shows the elegance and efficiency of the Power Rule in calculus.
To apply the Power Rule:
  • Identify the exponent \( n \) of the term.
  • Multiply the entire term by \( n \) and reduce the exponent by one.
Leveraging the simplicity of the Power Rule is crucial in differentiating complex functions quickly and accurately.

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