2.3 Q-4

Question Statement

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

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Background and Explanation

To solve this problem, we will use the quotient rule for differentiation. The quotient rule is applied when differentiating a function that is the ratio of two functions. The rule states: $\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$

Here, we have:

• $u(x) = 2x - 3$
• $v(x) = 2x + 1$

We will differentiate both the numerator and the denominator and then apply the quotient rule.

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Solution

Let the function be: $Y = \frac{2x - 3}{2x + 1}$

Now, differentiate both sides with respect to $x$ using the quotient rule:

1. Differentiate the numerator ($u(x) = 2x - 3$): $u'(x) = 2$

2. Differentiate the denominator ($v(x) = 2x + 1$): $v'(x) = 2$

3. Apply the quotient rule: Using the quotient rule formula: $\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$

   Substitute the values for $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$: $\frac{dy}{dx} = \frac{(2x + 1) \cdot 2 - (2x - 3) \cdot 2}{(2x + 1)^2}$

4. Simplify the expression: First, distribute the constants in the numerator: $\frac{dy}{dx} = \frac{2(2x + 1) - 2(2x - 3)}{(2x + 1)^2}$

   Now simplify the numerator: $\frac{dy}{dx} = \frac{4x + 2 - 4x + 6}{(2x + 1)^2}$

   Combine like terms: $\frac{dy}{dx} = \frac{8}{(2x + 1)^2}$

Thus, the derivative of the given function is: $\frac{dy}{dx} = \frac{8}{(2x + 1)^2}$

Final Answer: $\frac{dy}{dx} = \frac{8}{(2x + 1)^2}$

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Key Formulas or Methods Used

• Quotient Rule:
  If you have a function in the form $\frac{u(x)}{v(x)}$, the derivative is: $\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$

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Summary of Steps

1. Identify the function:
   $Y = \frac{2x - 3}{2x + 1}$

2. Differentiate the numerator and denominator:

   • $u'(x) = 2$
   • $v'(x) = 2$

3. Apply the quotient rule: $\frac{dy}{dx} = \frac{(2x + 1) \cdot 2 - (2x - 3) \cdot 2}{(2x + 1)^2}$

4. Simplify the expression: $\frac{dy}{dx} = \frac{8}{(2x + 1)^2}$

5. Final Answer: $\frac{dy}{dx} = \frac{8}{(2x + 1)^2}$