Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 18
Find \(y^{\prime}(1)\) $$ y=\frac{x^{3 / 2}+2}{x} $$
Short Answer
Expert verified
The value of \( y'(1) \) is \(-\frac{3}{2}\).
Step by step solution
01
Understand the given function
The function given is \( y = \frac{x^{3/2} + 2}{x} \). It is a rational function and can be simplified for differentiation.
02
Rewrite the function
Rewrite the function \( y = \frac{x^{3/2} + 2}{x} \) as \( y = x^{3/2 - 1} + \frac{2}{x} \). This becomes \( y = x^{1/2} + 2x^{-1} \).
03
Differentiate the function
Differentiate each term separately. For \( y = x^{1/2} + 2x^{-1} \), use the power rule. The derivative of \( x^{1/2} \) is \( \frac{1}{2}x^{-1/2} \) and the derivative of \( 2x^{-1} \) is \( -2x^{-2} \). So, \( y' = \frac{1}{2}x^{-1/2} - 2x^{-2} \).
04
Evaluate the derivative at x = 1
Substitute \( x = 1 \) into the derivative to find \( y'(1) \). We have \( y'(1) = \frac{1}{2}(1)^{-1/2} - 2(1)^{-2} \). Simplify this to \( y'(1) = \frac{1}{2} - 2 = -\frac{3}{2} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Function
A rational function is a function that can be expressed as the ratio of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. Rational functions can sometimes involve variables in their exponents or other complex forms.
For example, in the original function \( y = \frac{x^{3/2} + 2}{x} \):
For example, in the original function \( y = \frac{x^{3/2} + 2}{x} \):
- "\( x^{3/2} + 2 \)" is the numerator.
- "\( x \)" is the denominator.
Power Rule
The power rule is a useful tool in calculus for differentiating functions where the variable is raised to a power. It's an essential rule to grasp as it allows you to find the derivative of power functions quickly. The power rule states:\[ \frac{d}{dx} \left( x^n \right) = nx^{n-1}\]Here's how it works:
- Take the exponent \( n \) and multiply it by the coefficient.
- Then decrease the exponent by one to get the new power.
- The exponent here is \( \frac{1}{2} \).
- Applying the rule gives: \( \frac{1}{2}x^{-1/2} \).
Derivative Evaluation
Evaluating the derivative at a specific point involves substituting a given value into the derivative function. It's a practical way of finding the rate of change of a function at a specific point. Here, after differentiating the expression, you substitute \( x = 1 \) into the derivative.Consider the function given: the derived function is \( y' = \frac{1}{2}x^{-1/2} - 2x^{-2} \).
- Substitute \( x = 1 \): \( y'(1) = \frac{1}{2}(1)^{-1/2} - 2(1)^{-2} \).
- This simplifies to \( y'(1) = \frac{1}{2} - 2 \).
- Finally, calculate the result to find \( y'(1) = -\frac{3}{2} \).
Simplifying Expressions
Simplifying expressions is a fundamental skill necessary for working with complex functions. Simplification aids in making the differentiation process more manageable. In the given exercise, the transformation from a hard-to-differentiate form to an easier one is crucial.Taking the original function \( y = \frac{x^{3/2} + 2}{x} \), you begin by rewriting it:
- The term \( x^{3/2} \) over \( x \) simplifies to \( x^{3/2 - 1} \) which equals \( x^{1/2} \).
- Similarly, \( \frac{2}{x} \) becomes \( 2x^{-1} \).
