Question

If \[ y = (x - 1)(x + 2)(x^2 + 5)(x^4 + 8), \] then \[ \lim\limits_{x \to -1} \left( \frac{dy}{dx} \right) = ? \]

• A. \( -30 \)
• B. \( 30 \)
• C. \( 52 \)
• D. \( -52 \)

Hint:

For product differentiation, use the chain rule and logarithmic differentiation for ease of computation.

Answer 1

The Correct Option is B

Step 1: Differentiate the given function Given function: \[ y = (x
- 1)(x + 2)(x^2 + 5)(x^4 + 8). \] Using logarithmic differentiation: \[ \frac{dy}{dx} = (x + 2)(x^2 + 5)(x^4 + 8) \cdot \frac{d}{dx} (x
- 1) +\]\[ (x
- 1)(x^2 + 5)(x^4 + 8) \cdot \frac{d}{dx} (x + 2) + (x
- 1)(x + 2)(x^4 + 8) \cdot \frac{d}{dx} (x^2 + 5) + (x
- 1)(x + 2)(x^2 + 5) \cdot \frac{d}{dx} (x^4 + 8). \] Step 2: Compute derivatives \[ \frac{d}{dx} (x
- 1) = 1, \quad \frac{d}{dx} (x + 2) = 1, \quad \frac{d}{dx} (x^2 + 5) = 2x, \quad \frac{d}{dx} (x^4 + 8) = 4x^3. \] Step 3: Substitute \( x =
-1 \) \[ \frac{dy}{dx} \Big|_{x =
-1} = (
-1 + 2)( (
-1)^2 + 5)( (
-1)^4 + 8) (1) +\]\[ (
-1
- 1)((
-1)^2 + 5)((
-1)^4 + 8) (1) + (
-1
- 1)(
-1 + 2)((
-1)^4 + 8)(2(
-1)) + (
-1
- 1)(
-1 + 2)((
-1)^2 + 5)(4(
-1)^3). \] Simplifying each term and summing, we get: \[ \frac{dy}{dx} = 30. \]