Question

Compute the derivative of the given function. $$g(x)=\frac{x+7}{\sqrt{x}}$$

Short answer

The derivative is \( g'(x) = \frac{x - 7}{2x\sqrt{x}} \).

Step-by-step solution

Identify the Differentiation Rule

First, recognize that the function \( g(x) = \frac{x+7}{\sqrt{x}} \) is a quotient of two functions: \( u(x) = x + 7 \) and \( v(x) = \sqrt{x} \). We will use the Quotient Rule to find the derivative, which is given by the formula \( \frac{d}{dx}\bigg(\frac{u}{v}\bigg) = \frac{v \cdot u' - u \cdot v'}{v^2} \).

Find Derivatives of Numerator and Denominator

Next, we need the derivatives of \( u(x) = x + 7 \) and \( v(x) = \sqrt{x} \). The derivative of the numerator is \( u'(x) = 1 \). The derivative of the denominator using the power rule for \( x^{1/2} \) is \( v'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \).

Apply the Quotient Rule

Substitute \( u(x), v(x), u'(x), \) and \( v'(x) \) into the Quotient Rule formula: \( g'(x) = \frac{\sqrt{x} \cdot 1 - (x+7) \cdot \frac{1}{2\sqrt{x}}}{(\sqrt{x})^2} \).

Simplify the Derivative Expression

Simplify \( g'(x) = \frac{\sqrt{x} - \frac{x+7}{2\sqrt{x}}}{x} \). Multiply through by \( 2\sqrt{x} \) to eliminate the fraction: \( g'(x) = \frac{2x - (x+7)}{2x\sqrt{x}} \). Simplify further to obtain \( g'(x) = \frac{x - 7}{2x\sqrt{x}} \).

Express Final Answer in Simplest Form

Finally, write the simplified derivative as \( g'(x) = \frac{\sqrt{x} - 7x^{-1/2}}{2x} \), which equates to \( \frac{\sqrt{x} - \frac{7}{\sqrt{x}}}{2x} \).

Key concepts

Quotient Rule

When dealing with the differentiation of functions that are expressed as a quotient, such as a fraction with one function over another, the Quotient Rule is essential. It is particularly useful when a function can be expressed in the form \( \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable functions.
The Quotient Rule formula for differentiation is:

• \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

This formula might look complex, but its application becomes easier with practice:

• First, identify \( u \) and \( v \) from the function \( g(x) = \frac{x+7}{\sqrt{x}} \).
• Then find the derivatives of both functions, \( u'(x) \) and \( v'(x) \), before applying the rule.

Power Rule

The Power Rule is a fundamental tool in calculus used to differentiate functions of the form \( x^n \). It simplifies the process of finding the derivative, which is crucial when applying other rules such as the Quotient Rule.
Given a function \( f(x) = x^n \):

• The derivative is \( f'(x) = nx^{n-1} \).

In the problem \( v(x) = \sqrt{x} \), rewrite it as an exponent \( v(x) = x^{1/2} \). Then, apply the Power Rule as follows:

• The derivative \( v'(x) = \frac{1}{2}x^{-1/2} \), which is equivalent to \( \frac{1}{2\sqrt{x}} \).

This derivative of \( v(x) \) is essential in the Quotient Rule formula applied in this example.

Differentiation

Differentiation is the core operation in calculus that calculates the rate at which a function changes at any given point. It involves finding the derivative, which is a representation of this rate of change.
There are several rules and techniques for differentiation, including the Quotient Rule, Power Rule, and others.
Here's how differentiation is applied to solve the given function \( g(x) = \frac{x+7}{\sqrt{x}} \):

• Recognize the structure of \( g(x) \) as a quotient, and prepare to use the Quotient Rule.
• Differentiate the numerator and denominator separately: \( u'(x) = 1 \) and \( v'(x) = \frac{1}{2\sqrt{x}} \).

After obtaining these derivatives, substitute them into the Quotient Rule to find \( g'(x) \), the derivative of \( g(x) \).

Simplification

Simplification is the process of making mathematical expressions easier to understand or manipulate. In calculus, once the derivative is found, it often involves combining like terms and simplifying fractions.
The given derivative initially appears as a complex fraction, so simplifying it is crucial:

• Original derivative expression: \( g'(x) = \frac{\sqrt{x} - \frac{x+7}{2\sqrt{x}}}{x} \).
• To simplify, multiply through by \( 2\sqrt{x} \) to remove fractional parts.
• Resulting expression: \( g'(x) = \frac{2x - (x+7)}{2x\sqrt{x}} \).
• Further simplification yields: \( g'(x) = \frac{x - 7}{2x\sqrt{x}} \).

Finally, express the simplest form as \( g'(x) = \frac{\sqrt{x} - \frac{7}{\sqrt{x}}}{2x} \). Each step ensures the expression is as clear and manageable as possible.