Question

Find the derivative of the algebraic function. $$ f(x)=\frac{2 x+5}{\sqrt{x}} $$

Short answer

The derivative of the function \(f(x)=\frac{2x+5}{\sqrt{x}}\) is \(f'(x) = \frac{x^{1/2} - 5x^{-1/2}}{x}\).

Step-by-step solution

Identify the Numerator and the Denominator Functions

Firstly, identify the numerator function \(u\) and the denominator function \(v\). Here, \(u = 2x + 5\) and \(v = \sqrt{x}\) or \(v = x^{1/2}\).

Compute the Derivatives of Numerator and Denominator

Next, find the derivative of \(u\) and \(v\). The derivative of \(u = 2x + 5\) is \(u' = 2\) (since the derivative of a constant is 0). The derivative of \(v = x^{1/2}\) is \(v' = \frac{1}{2}x^{-1/2}\) using the power rule.

Apply the Quotient Rule

Substitute the functions (\(u\), \(v\)) and their derivatives (\(u'\), \(v'\)) into the quotient rule equation \(\frac{vu'-uv'}{v^2}\). Hence, the derivative \(f'(x)\) is \(\frac{(x^{1/2})(2) - (2x + 5)(\frac{1}{2}x^{-1/2})}{(x^{1/2})^2}\)

Simplify the Result

Work out the subtraction in the numerator and simplify the denominator which becomes \(f'(x) = \frac{2x^{1/2} - x^{-1/2}(2x + 5)}{x}\). This simplifies further to \(f'(x) = \frac{x^{1/2} - 5x^{-1/2}}{x}\).

Key concepts

Quotient Rule

When we're dealing with the derivative of a function that is the division of two differentiable functions, we employ the quotient rule. The quotient rule is a method for finding the derivative of a quotient of two functions. It states that for functions of the form \( \frac{u}{v} \), where both \( u \) and \( v \) are differentiable, the derivative, denoted as \( \frac{d}{dx}(\frac{u}{v}) \), is given by \( \frac{vu' - uv'}{v^2} \).

Here's a breakdown of the process:

• Firstly, identify the top function (the numerator) as \( u \) and the bottom function (the denominator) as \( v \).
• Secondly, find the derivatives of both \( u \) and \( v \) separately, denoted by \( u' \) and \( v' \) respectively.
• Finally, apply the quotient rule formula to compute the derivative of the entire function.

Understanding this process, and practicing it with various functions, will strengthen your ability to handle more complex differentiation problems involving ratios of functions.

Power Rule

The power rule is one of the fundamental tools in calculus for finding derivatives of functions. It states that if you have a function \( f(x) = x^n \), where \( n \) is any real number, the derivative of \( f(x) \) with respect to \( x \) is \( f'(x) = nx^{n-1} \). This rule greatly simplifies the process of differentiation because it provides a quick and straightforward method to find the derivative without the need for limits.

In our example, we see the power rule applied as follows: the derivative of \( x^{1/2} \) is \( \frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2} \). Similarly, for any term \( x^n \) within a larger expression, the power rule can be applied in the same manner to find its derivative, regardless of whether the expression is a simple monomial or part of a more complex function.

Simplifying Derivatives

Once we have applied rules such as the quotient or power rule and obtained a derivative expression, it's often necessary to simplify the result to make it easier to work with and understand. Simplification might involve combining like terms, reducing fractions, factoring, or canceling common factors.

In the provided exercise solution, after applying the quotient rule, we simplified the resulting expression. Here’s what that entails:

• Combining like terms in the numerator when possible.
• Dividing each term by the denominator to reduce the expression into a simpler form.

Ultimately, the goal of simplifying the derivative is to present it in the most reduced form. This makes it easier to evaluate the derivative at given points and to analyze the function's behavior. Following these steps ensures that the final expression for the derivative is both accurate and easy to interpret.