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Magazine Chapter 3: Problem 21
Differentiate the function. \(y=\frac{x^{2}+4 x+3}{\sqrt{x}}\)
Short Answer
Expert verified
The derivative is \( y' = \frac{x - 3}{x^{3/2}} \).
Step by step solution
01
Identify the Form of the Function
The function given is \( y = \frac{x^{2}+4x+3}{\sqrt{x}} \). This can be considered as a quotient of two functions: \( u(x) = x^2 + 4x + 3 \) and \( v(x) = \sqrt{x} \). We will use the quotient rule for differentiation.
02
Recall the Quotient Rule
The quotient rule states that if you have a function \( y = \frac{u}{v} \), then its derivative is given by \( y' = \frac{v \, u' - u \, v'}{v^2} \). We need to find \( u' \) and \( v' \) before applying this rule.
03
Differentiate the Numerator
The numerator function is \( u(x) = x^2 + 4x + 3 \). Its derivative, using standard differentiation rules, is \( u'(x) = 2x + 4 \).
04
Differentiate the Denominator
The denominator function is \( v(x) = \sqrt{x} = x^{1/2} \). Differentiating \( x^{1/2} \) with respect to \( x \) gives \( v'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \).
05
Apply the Quotient Rule
Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula:\[ y' = \frac{\sqrt{x} \, (2x + 4) - (x^2 + 4x + 3) \, \frac{1}{2\sqrt{x}}}{x}\]Simplify this expression to find the derivative.
06
Simplify the Expression
By further simplifying the expression from Step 5, we find:\[ y' = \frac{2x^{3/2} + 4x^{1/2} - \frac{x^2}{2} - 2x - \frac{3}{2}\sqrt{x}}{x}\]Combine like terms and simplify to get the final expression:\[ y' = \frac{3x^{-1/2} + 6x^{-3/2} - \frac{1}{2}x^{1/2} - 2x^{-1/2}}{x}\]
07
Final Simplification
Simplify the expression from the previous step further to arrive at:\[y' = \frac{x^2 \cdot (2x + 4) - \sqrt{x} \cdot (x^2 + 4x + 3)}{x^2}\]Simplify it finally to get:\[y' = \frac{x - 3}{x^{3/2}}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the quotient, or division, of two other functions. It's crucial in calculus and differentiation, particularly when dealing with fractions of functions. If you have a function expressed as \( y = \frac{u}{v} \),\ where \( u \) and \( v \) are differentiable functions of \( x \), the derivative \( y' \) can be calculated using the formula:
- \( y' = \frac{v \, u' - u \, v'}{v^2} \)
- Differentiate \( u \) with respect to \( x \), denoted as \( u' \).
- Differentiate \( v \) with respect to \( x \), denoted as \( v' \).
- Multiply \( v \) by \( u' \) and \( u \) by \( v' \), then subtract the second product from the first.
- Finally, divide the result by \( v^2 \), the square of the original denominator.
Numerator and Denominator Differentiation
When working with functions expressed as a quotient, like \( y = \frac{u(x)}{v(x)} \), it is essential to correctly differentiate the numerator \( u(x) \) and the denominator \( v(x) \) as part of applying the quotient rule effectively. Let's break it down for clarity:
- **Differentiate the Numerator:** Take the derivative of \( u(x) \). For example, for \( u(x) = x^2 + 4x + 3 \), the derivative is computed as \( u'(x) = 2x + 4 \). This involves applying basic differentiation rules, such as the power rule and the derivative of constants.
- **Differentiate the Denominator:** For the denominator function \( v(x) = \sqrt{x} = x^{1/2} \), you would find the derivative as \( v'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \). This often involves applying the power rule where the exponent is a fraction.
Simplification of Expressions
In calculus, simplifying expressions is a critical step which ensures that the derivative is presented in the most elegant and useful form. After applying the quotient rule, you are often left with a complex expression that needs to be simplified for easier interpretation and application.
- **Combine Like Terms:** Begin by identifying and combining like terms within the expression. This reduces the number of terms and makes the expression neater.
- **Reduce Fractions:** When your expression includes fractional terms, try to simplify by reducing these fractions wherever possible. Look for common factors in the numerator and the denominator.
- **Manage Exponents:** Use rules of exponents to simplify terms, especially those involving fractional powers. Switching between radical and exponent forms as needed can make calculations simpler.
