Problem 52 - Calculus Single Variable

Question

Let \(c=4, F(x)=1+3 x, G(x)=\sqrt{x}\) and \(H(x)=x /(x+5) .\) Calculate the requested derivative. $$ \left(\frac{d}{d x} F \circ\left(\frac{H}{G}\right)\right)(c) $$

Step-by-Step Solution

Answer

The derivative \(\left(\frac{d}{d x} F \circ\left(\frac{H}{G}\right)\right)(c)\) at \(c=4\) is \(\frac{1}{108}\).

Step 1: Identify Composite Function

The given expression is the derivative of the composition of functions: \(F\), \(H\), and \(G\). We need to find \(\frac{H}{G}(x)\) first, which is \(\frac{x/(x+5)}{\sqrt{x}}\). The composite function is \(F\left(\frac{H}{G}\right)(x) = 1 + 3\frac{x/(x+5)}{\sqrt{x}}\).

Step 2: Simplify the Inner Function

Simplify \(\frac{H}{G}(x)\). This gives: \[ \frac{x}{x+5} \cdot \frac{1}{\sqrt{x}} = \frac{\sqrt{x}}{x+5} \], because \(\frac{x}{\sqrt{x}} = \sqrt{x}\). Substitute into \(F(x)\): \[ F\left(\frac{H}{G}(x)\right) = 1 + 3 \cdot \frac{\sqrt{x}}{x+5} \].

Step 3: Differentiate the Composite Function

Use the chain rule to differentiate \(F\left(\frac{H}{G}(x)\right)\). The derivative \( (F(u))' = 3u' \), where \(u = \frac{\sqrt{x}}{x+5} \). First find \(u'\) using the quotient rule: \[ u' = \frac{(x+5)(\frac{1}{2\sqrt{x}}) - \sqrt{x}}{(x+5)^2} = \frac{(x+5)\cdot\frac{1}{2\sqrt{x}} - \sqrt{x}}{(x+5)^2} \].

Step 4: Evaluate the Derivative at \(c=4\)

Substitute \(x=4\) in \(u'(x)\): \[ u' = \frac{(4+5)\cdot \frac{1}{2\sqrt{4}} - \sqrt{4}}{(4+5)^2} = \frac{9 \cdot \frac{1}{4} - 2}{81} = \frac{\frac{9}{4} - 2}{81} = \frac{\frac{1}{4}}{81} = \frac{1}{324} \]. Multiply by 3: \(3u' = \frac{3}{324} = \frac{1}{108}\).

Step 5: Final Result

Thus, the derivative of the composition at \(x=4\) is \(\frac{1}{108}\).

Key Concepts

Chain Rule

In calculus, the chain rule helps us find the derivative of composite functions. Imagine you have two functions, say, \(f(x)\) and \(g(x)\), where the output of \(g(x)\) becomes the input for \(f(x)\). In such a scenario, if you want to differentiate the composite function \(f(g(x))\), you apply the chain rule.
The chain rule states:

• First, differentiate the outer function \(f\) with respect to the inner function \(g(x)\).
• Then, multiply this by the derivative of the inner function \(g\) with respect to \(x\).

This can be mathematically expressed as: \[\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\]It's like peeling layers of an onion, where you differentiate one layer at a time from the outside to the inside. It's especially important when dealing with nested functions as it allows us to break complex problems into simpler parts.

Quotient Rule

The quotient rule is a technique used to differentiate functions that are presented as a division of two expressions. Suppose you have a function \(h(x) = \frac{f(x)}{g(x)}\). To find the derivative of \(h(x)\), we apply the quotient rule, which is a special formula used in differentiation.
The formula for the quotient rule is:

• Take the derivative of the numerator \(f(x)\), multiply it by the denominator \(g(x)\).
• Then, subtract the product of the numerator \(f(x)\) and the derivative of the denominator \(g(x)\).
• Finally, divide all of this by the square of the denominator \(g(x)^2\).

In mathematical terms: \[\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]This rule is extremely handy when you're dealing with functions that are fractions, as it maintains precision and provides a systematic approach to finding derivatives.

Composite Function

A composite function, in simple terms, is a function that is made up of two or more functions. It's like taking the output of one function and feeding it as the input of another. In notation, if \(f\) and \(g\) are functions, the composite function \(f \circ g\) is defined by \(f(g(x))\).
To understand composite functions, think of them as boxed processes:

• First, the value of \(x\) goes through \(g(x)\) to get a new result.
• This result is then processed by \(f(x)\).

For example, in our original problem, \(F\) is applied to the output of \(\frac{H}{G}\), forming a composite of these functions. Recognizing and working with composite functions often involves the chain rule to dive into layers and simplify the differentiation process.

Differentiation

Differentiation is a fundamental concept in calculus, representing how a function changes as its input changes. It's all about finding the rate at which something varies, which is called the derivative. For any function \(f(x)\), the derivative, denoted \(f'(x)\) or \(\frac{df}{dx}\), tells us how \(f\) changes with a change in \(x\).
Here are a few key points about differentiation:

• It's the process of finding the derivative, and there are specific rules, like the chain rule and quotient rule, to find derivatives systematically.
• It is applied for calculating slopes of curves, rates of change, and optimization problems.
• In a simple graphical sense, finding a derivative at a point gives the slope of the tangent line to a function's graph at that point.

Understanding differentiation allows us to predict and model real-world behaviors, making it an invaluable tool in mathematics, physics, engineering, and beyond.