2.3 Q-9

Question Statement

Differentiate the following expression with respect to $x$:
$y = \frac{x^2 + 1}{x^2 - 3}$

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Background and Explanation

This problem involves differentiation of a rational function, which requires the quotient rule. The quotient rule is used when a function is expressed as a ratio of two other functions. Additionally, we need to know how to differentiate basic polynomial terms like $x^2$.

Key Differentiation Rules:

1. Quotient Rule:
   If $y = \frac{u}{v}$, then:
   $\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$
2. Differentiation of $x^n$:
   $\frac{d}{dx}(x^n) = n \cdot x^{n-1}$

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Solution

Step 1: Identify the numerator and denominator

The given function is:
$y = \frac{x^2 + 1}{x^2 - 3}$
Here:

• Numerator: $u = x^2 + 1$
• Denominator: $v = x^2 - 3$

Step 2: Apply the quotient rule

Using the quotient rule:
$\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$

a) Differentiate $u = x^2 + 1$:

$\frac{du}{dx} = 2x$

b) Differentiate $v = x^2 - 3$:

$\frac{dv}{dx} = 2x$

Step 3: Substitute into the quotient rule

Substitute $u$, $v$, and their derivatives into the formula: $\frac{dy}{dx} = \frac{(x^2 - 3)(2x) - (x^2 + 1)(2x)}{(x^2 - 3)^2}$

Step 4: Simplify the numerator

Expand both terms in the numerator:

1. $(x^2 - 3)(2x) = 2x(x^2 - 3) = 2x^3 - 6x$
2. $(x^2 + 1)(2x) = 2x(x^2 + 1) = 2x^3 + 2x$

Subtract these terms: $\text{Numerator} = \left(2x^3 - 6x\right) - \left(2x^3 + 2x\right)$
$\text{Numerator} = 2x^3 - 6x - 2x^3 - 2x$
$\text{Numerator} = -8x$

Step 5: Write the final expression

The derivative is: $\frac{dy}{dx} = \frac{-8x}{(x^2 - 3)^2}$

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Key Formulas or Methods Used

1. Quotient Rule:
   $\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$

2. Differentiation of polynomials:
   $\frac{d}{dx}(x^n) = n \cdot x^{n-1}$

3. Simplification of algebraic expressions.

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Summary of Steps

1. Identify $u$ (numerator) and $v$ (denominator).
2. Differentiate $u = x^2 + 1$ and $v = x^2 - 3$:
   • $\frac{du}{dx} = 2x$
   • $\frac{dv}{dx} = 2x$
3. Apply the quotient rule: $\frac{dy}{dx} = \frac{(x^2 - 3)(2x) - (x^2 + 1)(2x)}{(x^2 - 3)^2}$
4. Simplify the numerator to: $-8x$
5. Write the final expression: $\frac{dy}{dx} = \frac{-8x}{(x^2 - 3)^2}$