Problem 11 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ F(x)=\frac{\sqrt{x^{2}-1}}{x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{x^2 \sqrt{x^2 - 1}} \).

Step 1: Understand the Problem

The task is to find the derivative of the function \( F(x) = \frac{\sqrt{x^2 - 1}}{x} \). This involves using the quotient rule and the chain rule.

Step 2: Apply the Quotient Rule

Recall the quotient rule: If \( F(x) = \frac{g(x)}{h(x)} \), then \( F'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \). For the given function, \( g(x) = \sqrt{x^2 - 1} \) and \( h(x) = x \).

Step 3: Differentiate the Numerator using the Chain Rule

Find the derivative of the numerator \( g(x) = \sqrt{x^2 - 1} \). Let \( u = x^2 - 1 \), then \( g(x) = u^{1/2} \). The derivative of \( u^{1/2} \) using the chain rule is \( \frac{1}{2} u^{-1/2} \cdot 2x \) which simplifies to \( \frac{x}{\sqrt{x^2 - 1}} \).

Step 4: Differentiate the Denominator

The derivative of the denominator \( h(x) = x \) is simply \( h'(x) = 1 \).

Step 5: Combine Using the Quotient Rule

Substitute the derivatives into the quotient rule formula: \( F'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \). This gives: \[ F'(x) = \frac{ \left( \frac{x}{\sqrt{x^2 - 1}} \right) x - \sqrt{x^2 - 1} \cdot 1 }{x^2} \].

Step 6: Simplify the Expression

Simplify the numerator: \[ x \left( \frac{x}{\sqrt{x^2 - 1}} \right) - \sqrt{x^2 - 1} = \frac{x^2}{\sqrt{x^2 - 1}} - \sqrt{x^2 - 1} \]. Combine the terms over a common denominator: \[ \frac{x^2 - (x^2 - 1)}{\sqrt{x^2 - 1}} = \frac{1}{\sqrt{x^2 - 1}} \]. So, \[ F'(x) = \frac{ \frac{1}{\sqrt{x^2 - 1}} }{x^2} = \frac{1}{ x^2 \sqrt{x^2 - 1} } \].

Key Concepts

Quotient Rule

To find the derivative of a function that is a ratio of two other functions, we use the quotient rule. If we have a function of the form: \[F(x) = \frac{g(x)}{h(x)}\] the quotient rule states the derivative \(F'(x)\) is: \[F'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}\]. First, identify the numerator function \(g(x)\) and the denominator function \(h(x)\). In our case, \(g(x) = \sqrt{x^2 - 1}\) and \(h(x) = x\). Next, find the derivatives of \(g(x)\) and \(h(x)\), which we'll do using the chain rule for \(g(x)\). Replace \(g'(x)\), \(h(x)\), \(g(x)\), and \(h'(x)\) into the formula to find \(F'(x)\).

Chain Rule

The chain rule is used to differentiate composite functions, where one function is inside another. Here we need it for the numerator \(g(x) = \sqrt{x^2 - 1}\). To apply the chain rule:
- Let \(u = x^2 - 1\).*
- Rewrite \(g(x)\) as \((u)^{1/2}\).
- Differentiate \( (u)^{1/2}\) with respect to \(u\), giving \( \frac{1}{2} u^{-1/2} \).
- Then multiply by the derivative of \(u\) with respect to \(x\), which is \(2x\).
So, the derivative of \(g(x)\) is: \[ \frac{1}{2} (x^2 - 1)^{-1/2} \times 2x = \frac{x}{\sqrt{x^2 - 1}}\]. Now we've found \(g'(x)\). For \(h(x) = x\), the derivative \(h'(x)\) is simply \(1\).

Simplification of Expressions

After applying the quotient rule, you usually end up with a complex expression that requires simplification. Starting from: \[F'(x) = \frac{ \frac{x}{\sqrt{x^2 - 1}} \times x - \sqrt{x^2 - 1} \times 1 }{x^2}\],
we first simplify the numerator: \[ \frac{x^2}{\sqrt{x^2 - 1}} - \sqrt{x^2 - 1}\].
Combine the terms over a common denominator to: \[ \frac{x^2 - (x^2 - 1)}{\sqrt{x^2 - 1}} \].
Simplify the expression inside the numerator to get: \[ \frac{1}{\sqrt{x^2 - 1}}\].
So our final simplified derivative: \[F'(x) = \frac{1}{x^2 \sqrt{x^2 - 1}}\]. Simplification is crucial as it makes the final expression much clearer and easier to understand.