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Chapter 1: Problem 19

Find the derivative of \(f(x)\) at the designated value of \(x .\) $$f(x)=\frac{1}{x} \text { at } x=\frac{2}{3}$$

Short Answer

Expert verified
-\frac{9}{4}

Step by step solution

01

- Define the function

Given the function is \[ f(x) = \frac{1}{x} \]
02

- Find the derivative of the function

Use the power rule for derivatives. Since \[ f(x) = x^{-1} \], the derivative \[ f'(x) = \frac{d}{dx}(x^{-1}) \] is \[ f'(x) = -x^{-2} = -\frac{1}{x^2} \]
03

- Evaluate the derivative at the given x value

Substitute \[ x = \frac{2}{3} \] into the derivative. \[ f'\left(\frac{2}{3}\right) = -\frac{1}{(\frac{2}{3})^2} = -\frac{1}{\frac{4}{9}} = -\frac{9}{4} \]

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Key Concepts

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

Power Rule
The power rule is a fundamental principle in differentiation used to find the derivative of functions of the form \( x^n \). This rule states that if you have a function \( f(x) = x^n \), then its derivative is \( f'(x) = nx^{n-1} \). Here's a step-by-step guide to using the power rule:
  • First, identify the exponent \( n \) in the function \( x^n \).
  • Next, multiply the function by this exponent, \( n \).
  • Finally, decrease the exponent by one to obtain the derivative.
In our example, \( f(x) = \frac{1}{x} \), we first rewrite it using a negative exponent: \( f(x) = x^{-1} \). Applying the power rule:
\[ f'(x) = -x^{-2} = -\frac{1}{x^2} \]
Differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents how a function changes as its input changes. There are various techniques, rules, and formulas used in differentiation, but they all aim to measure the rate at which a dependent variable changes with respect to an independent variable.
It helps in finding the slope of the function at any point. The steps for differentiation are:
  • Identify the function you need to differentiate.
  • Apply the appropriate differentiation rule or formula.
  • Simplify the resulting expression to find the derivative.
In our case, we have:
  • The function is \( f(x) = \frac{1}{x} \)
  • Rewrite it as \( f(x) = x^{-1} \)
  • We apply the power rule to find its derivative \( f'(x) = -x^{-2} \).
Evaluation of Derivatives
Once we have found the derivative of a function, the next step usually involves evaluating this derivative at a specific point. This process is simple and involves substituting the given value of \( x \) into the derivative function to find the slope of the original function at that point. Here鈥檚 how to do it:
  • Take the derivative function, which in our case is \( f'(x) = -\frac{1}{x^2} \).
  • Substitute the given value of \( x \) into this derivative.
  • Simplify the resulting expression to get the final value.
For the given problem, we need to evaluate the derivative at \( x = \frac{2}{3} \):
  • Substitute \( x = \frac{2}{3} \) into \( f'(x) \): \[ f'\bigg( \frac{2}{3} \bigg) = -\frac{1}{ ( \frac{2}{3} )^2 } \]
  • Simplify inside the parenthesis: \[ -\frac{1}{ \frac{4}{9} } \]
  • Invert and multiply: \[ -\frac{9}{4} \]
Thus, the slope of the function \( f(x) \) at \( x = \frac{2}{3} \) is \( -\frac{9}{4} \).

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Most popular questions from this chapter

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x & \text { for } x \neq 1 \\ 2 & \text { for } x=1 \end{array}\right.$$

Suppose that 5 mg of a drug is injected into the bloodstream. Let \(f(t)\) be the amount present in the bloodstream after \(t\) hours. Interpret \(f(3)=2\) and \(f^{\prime}(3)=-.5 .\) Estimate the number of milligrams of the drug in the bloodstream after \(3 \frac{1}{2}\) hours.

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Function The revenue from producing (and selling) \(x\) units of a product is given by \(R(x)=3 x-.01 x^{2}\) dollars. (a) Find the marginal revenue at a production level of \(20 .\) (b) Find the production levels where the revenue is 200 dollars.

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=x^{2}$$
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