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Chapter 3: Problem 26

Differentiate the functions. $$y=\sqrt{\frac{x+3}{x^{2}+1}}$$

Short Answer

Expert verified
Final differentiated expression: \[ y' = \frac{(x^2+1) - 2x(x+3)}{2\sqrt{x+3}(x^2+1))} \]

Step by step solution

01

Rewrite the Function

Rewrite the function using exponent notation for the square root. This helps simplify the differentiation process. The function becomes: \[ y = \frac{(x+3)^{1/2}}{(x^2+1)^{1/2}} \]
02

Apply the Quotient Rule

The Quotient Rule states that for a function \( y = \frac{u}{v} \), the derivative \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = (x+3)^{1/2} \) and \( v = (x^2+1)^{1/2} \).
03

Differentiate the Numerator

Find the derivative of \( u = (x+3)^{1/2} \) using the Chain Rule: \[ u' = \frac{1}{2}(x+3)^{-1/2} \cdot 1 = \frac{1}{2\sqrt{x+3}} \]
04

Differentiate the Denominator

Find the derivative of \( v = (x^2+1)^{1/2} \) using the Chain Rule: \[ v' = \frac{1}{2}(x^2+1)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{x^2+1}} \]
05

Substitute into the Quotient Rule

Substitute \( u, u', v, \) and \( v' \) into the Quotient Rule formula: \[ y' = \frac{\left( \frac{1}{2\sqrt{x+3}} \right)(\sqrt{x^2+1}) - (\sqrt{x+3})(\frac{x}{\sqrt{x^2+1}})}{(\sqrt{x^2+1})^2} \]
06

Simplify the Expression

Simplify the numerator and denominator: \[ y' = \frac{\sqrt{x^2+1}}{2\sqrt{x+3} \cdot \sqrt{x^2+1}} - \frac{x\sqrt{x+3}}{(x^2+1)^{3/2}} \rightarrow \frac{1}{2\sqrt{x+3}} - \frac{x\sqrt{x+3}}{(x^2+1)} \]
07

Combine Terms

Combine the terms to get the final differentiated expression: \[ y' = \frac{(x^2+1) - 2x(x+3)}{2\sqrt{x+3}(x^2+1))} \]

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Key Concepts

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

Calculus
Calculus is the branch of mathematics focusing on change. It includes two main concepts: differentiation (finding rates of change) and integration (finding areas under curves). Differentiation, as we've seen, is essential for understanding how functions change at any point. The given problem used differentiation techniques to find the rate of change of a more complex function. Mastery of these basic calculus principles is foundational for advanced studies in mathematics and other fields like physics and engineering.

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Most popular questions from this chapter

Compute \(\left.\frac{d y}{d t}\right|_{t=t_{0}}\). $$y=\sqrt{x+1}, x=\sqrt{t+1}, t_{0}=0$$

A function \(h(x)\) is defined in terms of a differentiable \(f(x) .\) Find an expression for \(h^{\prime}(x)\). $$h(x)=\left(x^{2}+2 x-1\right) f(x)$$

A function \(h(x)\) is defined in terms of a differentiable \(f(x) .\) Find an expression for \(h^{\prime}(x).\) $$h(x)=f(f(x))$$

A closed rectangular box is to be constructed with one side 1 meter long. The material for the top costs \(\$ 20\) per square meter, and the material for the sides and bottom costs \(\$ 10\) per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of \(\$ 240\) for materials.

If \(f(x)\) and \(g(x)\) are differentiable functions such that \(f(2)=f^{\prime}(2)=3, g(2)=3,\) and \(g^{\prime}(2)=\frac{1}{3},\) compute the following derivatives: $$\left.\frac{d}{d x}\left[(f(x))^{2}\right]\right|_{x=2}$$
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