Q103 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics

If $y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)$, then $\lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)=$

  • A
    -30
  • Correct answer
    B
    30
  • C
    52
  • D
    -52

Solution

$y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)$ $\begin{aligned} & \frac{d y}{d x}=(x+2)\left(x^2+5\right)\left(x^4+8\right)+(x-1)\left(x^2+5\right)\left(x^4+8\right) \\ & +(x-1)(x+2)(2 x)\left(x^4+8\right)+(x-1)(x+2)\left(x^2+5\right)\left(4 x^3\right) \\ & \Rightarrow \quad \lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)=(1)(6)(9)+(-2)(6)(9) \\ & \quad+(-2)(1)(-2)(9)+(-2)(1)(6)(-4) \\ & =54-108+36+48=30 .\end{aligned}$

Hence, the correct answer is option: 'B'.

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