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Chapter 3: Problem 21

\(3-26\) Differentiate. $$f(t)=\frac{2 t}{2+\sqrt{t}}$$

Short Answer

Expert verified
The derivative is \( f'(t) = \frac{4+\sqrt{t}}{(2+\sqrt{t})^2} \).

Step by step solution

01

Apply Quotient Rule

To differentiate the function \( f(t) = \frac{2t}{2+\sqrt{t}} \), we'll use the quotient rule, which states that if \( u(t) = \frac{v(t)}{w(t)} \), then \( u'(t) = \frac{v'(t)w(t) - v(t)w'(t)}{(w(t))^2} \). Here, \( v(t) = 2t \) and \( w(t) = 2 + \sqrt{t} \).
02

Differentiate the Numerator

Differentiate the numerator \( v(t) = 2t \). The derivative \( v'(t) = \frac{d}{dt}(2t) = 2 \).
03

Differentiate the Denominator

Differentiate the denominator \( w(t) = 2 + \sqrt{t} \). First, note that \( \sqrt{t} = t^{1/2} \). The derivative \( w'(t) = \frac{d}{dt}(2) + \frac{d}{dt}(t^{1/2}) = 0 + \frac{1}{2}t^{-1/2} = \frac{1}{2\sqrt{t}} \).
04

Substitute into the Quotient Rule

Substitute \( v(t) \), \( v'(t) \), \( w(t) \), and \( w'(t) \) into the quotient rule formula: \[ f'(t) = \frac{2(2 + \sqrt{t}) - 2t\left(\frac{1}{2\sqrt{t}}\right)}{(2 + \sqrt{t})^2} \].
05

Simplify the Expression

Simplify the expression: First simplify the numerator \( 2(2 + \sqrt{t}) - 2t\left(\frac{1}{2\sqrt{t}}\right) = 4 + 2\sqrt{t} - \frac{2t}{2\sqrt{t}} = 4 + 2\sqrt{t} - \frac{t}{\sqrt{t}} = 4 + 2\sqrt{t} - \sqrt{t} = 4 + \sqrt{t} \). Now the derivative can be written as \[ f'(t) = \frac{4+\sqrt{t}}{(2+\sqrt{t})^2} \].

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Key Concepts

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

Quotient Rule
When differentiating functions that are written as a quotient, or fraction, we use the quotient rule. This rule helps us find the derivative of a division of two functions. Think of it like having a sandwich where each layer plays a role in getting to the derivative.
  • The quotient rule is a formula which states that if you have a function like \( u(t) = \frac{v(t)}{w(t)} \), then its derivative is: \[ u'(t) = \frac{v'(t)w(t) - v(t)w'(t)}{w(t)^2}\]
  • It's important because it gives the exact rate of change of the function without needing to break it into smaller parts.
To apply the quotient rule, identify which parts of your function correspond to \( v(t) \) and \( w(t) \). Then, find their respective derivatives, \( v'(t) \) and \( w'(t) \). Finally, put it all into the quotient rule formula. Remember, practice makes perfect. Work through some examples to get the hang of substituting into the formula and simplifying the results.
Derivative of a Function
A derivative represents the rate of change of a function with respect to one of its variables. It鈥檚 like measuring how fast something is changing right at that very moment. For straightforward functions, calculating a derivative can be simple. However, more complex functions, like quotients, involve different rules, such as the quotient rule or the chain rule. When calculating the derivative of a function, like \( f(t) = \frac{2t}{2+\sqrt{t}} \), follow these steps:
  • Step 1: Identify the function terms - here, identify \( v(t) = 2t \) and \( w(t) = 2 + \sqrt{t} \).
  • Step 2: Compute each part's derivative individually to find \( v'(t) \) and \( w'(t) \).
  • Step 3: Substitute them back into the quotient rule formula.
Understanding each step in finding a derivative will strengthen your calculus skills. This foundational tool is essential in physics, engineering, and other scientific fields, helping you solve real-world problems.
Simplifying Expressions
After finding the derivative using the quotient rule, your last task is simplifying the resulting expression. Simplification is key in mathematics to make expressions easier to understand and work with. It helps in reducing complex fractions, eliminating redundancies, and presenting cleaner answers.Take the derivative from the exercise:\[f'(t) = \frac{2(2 + \sqrt{t}) - 2t\left(\frac{1}{2\sqrt{t}}\right)}{(2 + \sqrt{t})^2}\]Steps to Simplify:
  • Step 1: Simplify the numerator. Clean up terms: \( 2(2 + \sqrt{t}) - \frac{2t}{2\sqrt{t}} \rightarrow 4 + 2\sqrt{t} - \sqrt{t} \rightarrow 4 + \sqrt{t} \)
  • Step 2: Write the simplified derivative: \( f'(t) = \frac{4+\sqrt{t}}{(2+\sqrt{t})^2} \)
  • Step 3: Double-check each term to ensure no further simplification is possible.
Simplifying expressions not only makes them prettier but also provides clearer insights and interpretations, especially when it comes to real-world applications.

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Most popular questions from this chapter

The rate of change of atmospheric pressure \(\mathrm{P}\) with respect to altitude h is proportional to \(\mathrm{P}\) , provided that the temperature is constant. At \(15^{\circ} \mathrm{C}\) the pressure is 101.3 \(\mathrm{kPa}\) at sea level and 87.14 \(\mathrm{kPa}\) at \(\mathrm{h}=1000 \mathrm{m}\) (a) What is the pressure at an altitude of 3000 \(\mathrm{m}\) ? (b) What is the pressure at the top of Mount McKinley, at an altitude of 6187 \(\mathrm{m}\) ?

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