2.3 Q-5

Question Statement

Differentiate the following product of terms with respect to $x$:
$Y = (x-5)(3-x)$

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Background and Explanation

This question involves finding the derivative of a product of two terms. To solve this, we use the product rule of differentiation, which states:
$\frac{d}{dx} \left[ u \cdot v \right] = u \frac{dv}{dx} + v \frac{du}{dx}$
Here, $u$ and $v$ are functions of $x$. This rule ensures that both the variation of $u$ and $v$ are accounted for when differentiating their product.

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Solution

Let the given function be: $Y = (x - 5)(3 - x)$

Step 1: Identify $u$ and $v$

• Let $u = (x - 5)$
• Let $v = (3 - x)$

Step 2: Differentiate $u$ and $v$

• The derivative of $u = (x - 5)$ is:
  $\frac{du}{dx} = 1$
• The derivative of $v = (3 - x)$ is:
  $\frac{dv}{dx} = -1$

Step 3: Apply the product rule

Using the product rule:
$\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$

Substitute the values of $u$, $v$, $\frac{du}{dx}$, and $\frac{dv}{dx}$: $\frac{dy}{dx} = (x - 5)(-1) + (3 - x)(1)$

Step 4: Simplify the expression

Expand and combine terms:

• $(x - 5)(-1) = -x + 5$
• $(3 - x)(1) = 3 - x$

Adding these: $\frac{dy}{dx} = -x + 5 + 3 - x$ $\frac{dy}{dx} = -2x + 8$

Final Answer:
$\frac{dy}{dx} = -2x + 8$

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Key Formulas or Methods Used

• Product Rule:
  $\frac{d}{dx} \left[ u \cdot v \right] = u \frac{dv}{dx} + v \frac{du}{dx}$

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Summary of Steps

1. Identify the terms: $u = (x-5)$ and $v = (3-x)$.
2. Differentiate each term:
   • $\frac{du}{dx} = 1$
   • $\frac{dv}{dx} = -1$
3. Apply the product rule:
   $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$
4. Simplify the expression:
   $\frac{dy}{dx} = -2x + 8$