2.3 Q-3

Question Statement

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

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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 you are differentiating a function that is the ratio of two functions, such as $\frac{u(x)}{v(x)}$. The quotient 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, $u(x) = a + x$ and $v(x) = a - x$, so we need to differentiate both the numerator and the denominator.

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Solution

Let the function be: $Y = \frac{a + x}{a - x}$

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

1. Differentiate the numerator ($u(x) = a + x$): $u'(x) = 1$

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

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

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

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

   Combine like terms in the numerator: $\frac{dy}{dx} = \frac{2a}{(a - x)^2}$

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

Final Answer: $\frac{dy}{dx} = \frac{2a}{(a - x)^2}$

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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:
   $Y = \frac{a + x}{a - x}$

2. Apply the quotient rule:

   • Differentiate the numerator $u(x) = a + x$ to get $u'(x) = 1$
   • Differentiate the denominator $v(x) = a - x$ to get $v'(x) = -1$

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

4. Simplify the expression: $\frac{dy}{dx} = \frac{2a}{(a - x)^2}$

5. Final Answer: $\frac{dy}{dx} = \frac{2a}{(a - x)^2}$