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Magazine Chapter 3: Problem 14
Find the first derivative. $$ P(x)=\left(x+x^{-1}\right)^{2} $$
Short Answer
Expert verified
The first derivative of \( P(x) = (x + x^{-1})^2 \) is \( 2x - 2x^{-3} \).
Step by step solution
01
Identify the Function Structure
The given function is \( P(x) = (x + x^{-1})^2 \). This suggests a composition function of the form \( u^2 \) where \( u = x + x^{-1} \). To find the derivative, we'll need to use the chain rule.
02
Apply the Chain Rule
To differentiate \( P(x) = u^2 \) with respect to \( x \), we apply the chain rule. The derivative \( \frac{d}{dx}[u^2] = 2u \cdot \frac{du}{dx} \), where \( u = x + x^{-1} \). Next, we need to find \( \frac{du}{dx} \).
03
Differentiate Inside the Parenthesis
Let \( u = x + x^{-1} \). Differentiate to get \( \frac{du}{dx} = 1 - x^{-2} \) or \( 1 - \frac{1}{x^2} \). We use the power rule for \( x \) and the power rule combined with the chain rule for \( x^{-1} \).
04
Substitute Back
Substitute \( u = x + x^{-1} \) and \( \frac{du}{dx} = 1 - \frac{1}{x^2} \) into the chain rule expression: \( \frac{d}{dx}[P(x)] = 2(x + x^{-1})(1 - \frac{1}{x^2}) \).
05
Simplify the Expression
To simplify: distribute \( 2(x + x^{-1}) \) with \( (1 - \frac{1}{x^2}) \). This yields: \( 2(x + x^{-1}) - 2(x + x^{-1})\frac{1}{x^2} = 2x + 2x^{-1} - 2x^{-1} - 2x^{-3} \). The terms \( 2x^{-1} \) cancel out, resulting in: \( 2x - 2x^{-3} \).
06
Final Expression of the Derivative
The first derivative of \( P(x) \) is \( P'(x) = 2x - 2x^{-3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Chain Rule
In calculus, the chain rule is a fundamental principle that allows us to differentiate composite functions. Composite functions are functions within functions, like a nesting of operations. For our function \( P(x) = (x + x^{-1})^2 \), the outer function is \( u^2 \), and the inner function is \( u = x + x^{-1} \).
This means we can see our original problem as having two layers: the square and the expression inside it.
This means we can see our original problem as having two layers: the square and the expression inside it.
- To apply the chain rule, identify your "outer" function and your "inner" function.
- The chain rule formula is \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
- For our function, the derivative becomes \( 2u \cdot \frac{du}{dx} \).
Applying the Power Rule
The power rule is one of the most straightforward differentiation rules to apply. Simply put, if you have a function \( f(x) = x^n \), the power rule tells us \( f'(x) = nx^{n-1} \). This rule applies to each term we might encounter.
Let's revisit the differentiation of the component \( u = x + x^{-1} \):
Let's revisit the differentiation of the component \( u = x + x^{-1} \):
- For \( x \), using the power rule, the derivative is \( 1 \cdot x^{1-1} = 1 \).
- For \( x^{-1} \), the derivative is \( -1 \cdot x^{-2} = -x^{-2} \) since we apply \( -1x^{-2+1} \).
Fundamentals of Differentiation
Differentiation is a crucial concept in calculus that underpins many other principles, including the chain and power rules discussed earlier. At its most basic, differentiation involves finding the rate at which a function changes at any given point. This 'rate of change' forms the derivative.
For the function \( P(x) = (x + x^{-1})^2 \), differentiation helps us understand how changes in \( x \) influence changes in \( P(x) \).
For the function \( P(x) = (x + x^{-1})^2 \), differentiation helps us understand how changes in \( x \) influence changes in \( P(x) \).
- The goal is to break down functions to their simplest form to easily express how they're changing.
- Differentiation involves a systematic process of applying rules like the chain and power rules.
- Ultimately, carrying out these operations provides us with a functional derivative \( P'(x) = 2x - 2x^{-3} \).
