Problem 12 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ G(x)=\frac{4 x+6}{\sqrt{x^{2}+3 x+4}} $$

Step-by-Step Solution

Answer

\( G'(x) = \frac{3x + 7}{(x^2 + 3x + 4)^{3/2}} \)

Step 1 - Identify the function components

The given function is a quotient of two functions: the numerator is \(4x + 6\) and the denominator is \(\sqrt{x^2 + 3x + 4}\).

Step 2 - Apply the Quotient Rule

The quotient rule states that if you have a function \(\frac{u(x)}{v(x)}\), then the derivative is given by \(\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\). Here, \(u(x) = 4x + 6\) and \(v(x) = \sqrt{x^2 + 3x + 4}\).

Step 3 - Find the derivative of the numerator

Differentiate \(u(x) = 4x + 6\). The derivative is \(u'(x) = 4\).

Step 4 - Find the derivative of the denominator

First, rewrite the denominator \(v(x) = (x^2 + 3x + 4)^{1/2}\). Then use the chain rule to differentiate it. The chain rule gives \(v'(x) = \frac{1}{2}(x^2 + 3x + 4)^{-1/2} (2x + 3) = \frac{2x + 3}{2\sqrt{x^2 + 3x + 4}}\).

Step 5 - Apply the Quotient Rule Formula

Now that you have \(u'(x) = 4\), \(u(x) = 4x + 6\), \(v(x) = \sqrt{x^2 + 3x + 4}\), and \(v'(x) = \frac{2x + 3}{2\sqrt{x^2 + 3x + 4}}\), plug these into the quotient rule formula: \[ G'(x) = \frac{(4)(\sqrt{x^2 + 3x + 4}) - (4x + 6)\left(\frac{2x + 3}{2\sqrt{x^2 + 3x + 4}}\right)}{(\sqrt{x^2 + 3x + 4})^2} \]. Simplify the numerator: \[ 4\sqrt{x^2 + 3x + 4} - (4x + 6)\frac{2x + 3}{2\sqrt{x^2 + 3x + 4}} \]. Multiply to combine terms over a common denominator: \[ 4\sqrt{x^2 + 3x + 4} - \frac{(4x + 6)(2x + 3)}{2\sqrt{x^2 + 3x + 4}} \].

Step 6 - Simplify the expression

Combine like terms in the numerator: \[ \frac{8(x^2 + 3x + 4) - (4x + 6)(2x + 3)}{2\sqrt{x^2 + 3x + 4}(x^2 + 3x + 4)} \]. Further simplify the numerator: \[ 8(x^2 + 3x + 4) - (8x^2 + 18x + 18) = \frac{8x^2 + 24x + 32 - 8x^2 - 18x - 18}{2(x^2 + 3x + 4)^{3/2}} \]. Combine like terms: \[ \frac{6x + 14}{2(x^2 + 3x + 4)^{3/2}} = \frac{3x + 7}{(x^2 + 3x + 4)^{3/2}} \].

Key Concepts

Quotient Rule

When dealing with derivatives of functions that are represented as fractions, the Quotient Rule is a powerful tool. The rule states that if you have a function in the form \(\frac{u(x)}{v(x)}\), then the derivative of this function is found using the formula: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]. Breaking it down:

• \