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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differentiation

NDA 2018 · Mathematics · Differentiation

If y= ^ x^2 2 x , then what is dy dx at x=π equal to?

If y=\mathrmex2sin⁡2x\mathrm{y}=\mathrme^{\mathrm{x}^2} \sin 2 \mathrm{x}, then what is dydx\frac{\mathrm{dy}}{\mathrm{dx}} at x=π\mathrm{x}=\pi equal to?

Options

  1. A(1+π)\mathrmeπ2(1+\pi) \mathrme^{\pi^2}
  2. B2π\mathrmeπ22 \pi \mathrme^{\pi^2}
  3. C2\mathrmeπ22 \mathrme^{\pi^2}
  4. D\mathrmeπ2\mathrme^{\pi^2}

Correct answer

C. 2\mathrmeπ22 \mathrme^{\pi^2}

Step-by-step solution

y=ex2⋅sin⁡2xdydx=2⋅ex2⋅cos⁡2x+2xex2⋅sin⁡2x=2ex2(cos⁡2x+xsin⁡2x)dydx∣x=π=2eπ2(cos⁡2π+π⋅sin⁡2π)=2eπ2(1+0)=2⋅eπ2\begin{aligned} & y=e^{x^2} \cdot \sin 2 x \\ & \frac{d y}{d x}=2 \cdot e^{x^2} \cdot \cos 2 x+2 x e^{x^2} \cdot \sin 2 x \\ & =2 e^{x^2}(\cos 2 x+x \sin 2 x) \\ & \left.\frac{d y}{d x}\right|_{x=\pi}=2 e^{\pi^2}(\cos 2 \pi+\pi \cdot \sin 2 \pi) \\ & =2 e^{\pi^2}(1+0)=2 \cdot e^{\pi^2}\end{aligned}
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