Find The Derivative Of The Function.$y = \sqrt{\frac{x}{x+5}}$y^{\prime} =$

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Introduction

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In calculus, finding the derivative of a function is a crucial concept that helps us understand the rate of change of the function with respect to its input. In this article, we will focus on finding the derivative of a complex function, specifically the function $y = \sqrt{\frac{x}{x+5}}$. This function involves a square root and a fraction, making it a challenging problem to solve. However, with the right approach and techniques, we can break it down into manageable steps and find the derivative.

Understanding the Function

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Before we dive into finding the derivative, let's take a closer look at the function $y = \sqrt{\frac{x}{x+5}}$. This function involves a square root and a fraction, which can be rewritten as $y = \frac{\sqrt{x}}{\sqrt{x+5}}$. We can see that the function has two parts: the numerator and the denominator. The numerator is a square root of $x$, and the denominator is a square root of $x+5$.

Applying the Quotient Rule

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To find the derivative of this function, we can use the quotient rule, which states that if we have a function of the form $y = \frac{f(x)}{g(x)}$, then the derivative of $y$ is given by $y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}$. In our case, $f(x) = \sqrt{x}$ and $g(x) = \sqrt{x+5}$.

Finding the Derivative of the Numerator

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To find the derivative of the numerator, we can use the power rule, which states that if we have a function of the form $y = x^n$, then the derivative of $y$ is given by $y^{\prime} = nx^{n-1}$. In our case, $n = \frac{1}{2}$, so the derivative of the numerator is $f^{\prime}(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$.

Finding the Derivative of the Denominator

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To find the derivative of the denominator, we can use the chain rule, which states that if we have a function of the form $y = f(g(x))$, then the derivative of $y$ is given by $y^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x)$. In our case, $f(x) = \sqrt{x}$ and $g(x) = x+5$, so the derivative of the denominator is $g^{\prime}(x) = \frac{1}{2}(x+5)^{-\frac{1}{2}} = \frac{1}{2\sqrt{x+5}}$.

Applying the Quotient Rule

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Now that we have found the derivatives of the numerator and the denominator, we can apply the quotient rule to find the derivative of the function. Plugging in the values, we get:

$$y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}$$

$$y^{\prime} = \frac{\left(\frac{1}{2\sqrt{x}}\right)\left(\sqrt{x+5}\right) - \left(\sqrt{x}\right)\left(\frac{1}{2\sqrt{x+5}}\right)}{\left(\sqrt{x+5}\right)^2}$$

Simplifying the Expression

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To simplify the expression, we can start by multiplying out the terms in the numerator:

$$y^{\prime} = \frac{\frac{1}{2\sqrt{x}}\sqrt{x+5} - \frac{1}{2\sqrt{x+5}}\sqrt{x}}{\left(\sqrt{x+5}\right)^2}$$

Canceling Out Common Factors

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We can see that the numerator has a common factor of $\frac{1}{2\sqrt{x}}$ and $\frac{1}{2\sqrt{x+5}}$. We can cancel out these factors to simplify the expression:

$$y^{\prime} = \frac{\sqrt{x+5} - \sqrt{x}}{\left(\sqrt{x+5}\right)^2}$$

Rationalizing the Denominator

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To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator:

$$y^{\prime} = \frac{\sqrt{x+5} - \sqrt{x}}{\left(\sqrt{x+5}\right)^2} \cdot \frac{\sqrt{x+5} + \sqrt{x}}{\sqrt{x+5} + \sqrt{x}}$$

Simplifying the Expression

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After multiplying out the terms, we can simplify the expression:

$$y^{\prime} = \frac{\left(\sqrt{x+5}\right)^2 - \left(\sqrt{x}\right)^2}{\left(\sqrt{x+5}\right)^2\left(\sqrt{x+5} + \sqrt{x}\right)}$$

Canceling Out Common Factors

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We can see that the numerator has a common factor of $\left(\sqrt{x+5}\right)^2$ and $\left(\sqrt{x}\right)^2$. We can cancel out these factors to simplify the expression:

$$y^{\prime} = \frac{\left(\sqrt{x+5}\right)^2 - \left(\sqrt{x}\right)^2}{\left(\sqrt{x+5}\right)^2\left(\sqrt{x+5} + \sqrt{x}\right)}$$

Simplifying the Expression

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After canceling out the common factors, we can simplify the expression:

$$y^{\prime} = \frac{x+5 - x}{\left(\sqrt{x+5}\right)^2\left(\sqrt{x+5} + \sqrt{x}\right)}$$

Simplifying the Expression

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After simplifying the numerator, we can simplify the expression:

$$y^{\prime} = \frac{5}{\left(\sqrt{x+5}\right)^2\left(\sqrt{x+5} + \sqrt{x}\right)}$$

Simplifying the Expression

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After simplifying the denominator, we can simplify the expression:

$$y^{\prime} = \frac{5}{\left(x+5\right)\left(\sqrt{x+5} + \sqrt{x}\right)}$$

Conclusion

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In this article, we have found the derivative of the function $y = \sqrt{\frac{x}{x+5}}$. We used the quotient rule to find the derivative, and then simplified the expression to get the final answer. The derivative of the function is given by:

$$y^{\prime} = \frac{5}{\left(x+5\right)\left(\sqrt{x+5} + \sqrt{x}\right)}$$

This derivative can be used to find the rate of change of the function with respect to its input.

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Q: What is the quotient rule, and how is it used to find the derivative of a complex function?

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A: The quotient rule is a formula used to find the derivative of a function that is a quotient of two functions. It states that if we have a function of the form $y = \frac{f(x)}{g(x)}$, then the derivative of $y$ is given by $y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}$. This rule is used to find the derivative of a complex function by breaking it down into smaller parts and finding the derivatives of each part.

Q: How do I apply the quotient rule to find the derivative of a complex function?

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A: To apply the quotient rule, you need to identify the numerator and the denominator of the function. Then, you need to find the derivatives of the numerator and the denominator using the power rule and the chain rule. Finally, you can plug these derivatives into the quotient rule formula to find the derivative of the function.

Q: What is the power rule, and how is it used to find the derivative of a function?

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A: The power rule is a formula used to find the derivative of a function that is a power of a variable. It states that if we have a function of the form $y = x^n$, then the derivative of $y$ is given by $y^{\prime} = nx^{n-1}$. This rule is used to find the derivative of a function by multiplying the exponent by the variable and then subtracting 1 from the exponent.

Q: What is the chain rule, and how is it used to find the derivative of a function?

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A: The chain rule is a formula used to find the derivative of a function that is a composition of two or more functions. It states that if we have a function of the form $y = f(g(x))$, then the derivative of $y$ is given by $y^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x)$. This rule is used to find the derivative of a function by finding the derivatives of each part and then multiplying them together.

Q: How do I simplify the expression after applying the quotient rule?

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A: To simplify the expression after applying the quotient rule, you need to cancel out any common factors in the numerator and the denominator. You can also use algebraic manipulations such as multiplying out the terms and combining like terms to simplify the expression.

Q: What is the final answer for the derivative of the function $y = \sqrt{\frac{x}{x+5}}$?

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A: The final answer for the derivative of the function $y = \sqrt{\frac{x}{x+5}}$ is given by:

$$y^{\prime} = \frac{5}{\left(x+5\right)\left(\sqrt{x+5} + \sqrt{x}\right)}$$

This derivative can be used to find the rate of change of the function with respect to its input.

Q: Can I use the quotient rule to find the derivative of any function?

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A: Yes, you can use the quotient rule to find the derivative of any function that is a quotient of two functions. However, you need to make sure that the function is in the correct form, and you need to identify the numerator and the denominator correctly.

Q: What are some common mistakes to avoid when applying the quotient rule?

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A: Some common mistakes to avoid when applying the quotient rule include:

• Not identifying the numerator and the denominator correctly
• Not finding the derivatives of the numerator and the denominator correctly
• Not canceling out common factors in the numerator and the denominator
• Not simplifying the expression correctly

By avoiding these mistakes, you can ensure that you get the correct derivative of the function using the quotient rule.