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Chapter 2: Problem 22

Find \(f^{\prime}(x)\). \(f(x)=\frac{2}{x}-\frac{x}{2}\)

Short Answer

Expert verified
The derivative is \(f^{\text{'}}(x) = -\frac{2}{x^2} - \frac{1}{2} \).

Step by step solution

01

Identify the Components of the Function

The function given is the sum of two terms: \[ f(x) = \frac{2}{x} - \frac{x}{2} \]Each term will be differentiated separately.
02

Differentiate the First Term

The first term to differentiate is \[ \frac{2}{x} \ = 2x^{-1} \]Applying the power rule for differentiation, \( \frac{d}{dx} [2x^{-1}] = 2 \times (-1) x^{-2} = -\frac{2}{x^2} \).
03

Differentiate the Second Term

The second term to differentiate is \[ \frac{x}{2} \ = \frac{1}{2} x \]Applying the constant multiple rule, \( \frac{d}{dx} \bigg[\frac{1}{2} x\bigg] = \frac{1}{2} \).
04

Combine Both Derivatives

Combine the derivatives of both terms: \[ f^{\text{'}}(x) = -\frac{2}{x^2} - \frac{1}{2} \].

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Key Concepts

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

Power Rule Differentiation
To understand the power rule in differentiation, let鈥檚 start with its basic formula. The power rule states: \[ \frac{d}{dx}[x^n] = nx^{n-1} \]. This formula is applied whenever you're differentiating a term of the form \ (x^n)\. It simplifies the process of finding the derivative by following a few straightforward steps:
  • First, multiply the coefficient (if any) by the exponent.
  • Then, decrement the exponent by one.
In our exercise, the first term before differentiation was \( \frac{2}{x} = 2x^{-1} \). Using the power rule, the steps become clear: multiply 2 (the constant coefficient) by -1 (the old exponent), resulting in \(-2x^{-2}\). Simplifying, it becomes \(-\frac{2}{x^2} \).
Constant Multiple Rule
The constant multiple rule helps us differentiate terms with a constant multiplier. The rule is: \[ \frac{d}{dx} [c \bullet g(x)] = c \bullet \frac{d}{dx} [g(x)] \], where \( c \) is a constant and \( g(x) \) is a function of \( x \). For the given function \( f(x) = \frac{x}{2} = \frac{1}{2}x \), it's straightforward to use the constant multiple rule:
  • Identify the constant multiplier, which is \( \frac{1}{2} \) in this case.
  • Differentiate the remaining variable part of the term, \( x \).
  • The differentiation of \( x \) is simply 1.
Combining these steps gives \( \frac{1}{2} \times 1 = \frac{1}{2} \). So, the derivative of \( \frac{x}{2} \) is \( \frac{1}{2} \).
Combining Derivatives
After differentiating each term separately, the next step is to combine them. According to the linearity property of derivatives, if you have a function composed of multiple terms, you can find the derivative of each term individually and then combine them. For our function \( f(x) = \frac{2}{x} - \frac{x}{2} \), we already calculated:
  • The derivative of \( \frac{2}{x} \) is \( -\frac{2}{x^2} \)
  • The derivative of \( \frac{x}{2} \) is \( \frac{1}{2} \)
Combining these, we get: \[ f^{\text{'}}(x) = -\frac{2}{x^2} - \frac{1}{2} \]. There's nothing more to simplify, making this the final answer. Combining derivatives helps in breaking down complex functions into manageable parts, making calculations easier and more intuitive.

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Most popular questions from this chapter

Differentiate. $$ y=\sqrt{\frac{3 x-4}{5 x+3}} $$

For \(y=\sin x\), find a) \(\frac{d y}{d x}\). b) \(\frac{d^{2} y}{d x^{2}}\). c) \(\frac{d^{3} y}{d x^{3}}\). d) \(\frac{d^{4} y}{d x^{4}}\). e) \(\frac{d^{8} y}{d x^{8}}\). f) \(\frac{d^{10} y}{d x^{10}}\). g) \(\frac{d^{837} y}{d x^{837}}\).

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