2016 THREE(d)

Merit
Question
If y=exsin⁡xy = \dfrac{e^x}{\sin x}, show that dydx=y(1−cot⁡x)\dfrac{dy}{dx} = y(1 - \cot x).
Official Answer
y=exsin⁡xy=\dfrac{e^x}{\sin x}
dydx=sin⁡x⋅ex−excos⁡xsin⁡2x\dfrac{dy}{dx}=\dfrac{\sin x\cdot e^x-e^x\cos x}{\sin^2 x}
=sin⁡x⋅exsin⁡2x−ex⋅cos⁡xsin⁡2x=\dfrac{\sin x\cdot e^x}{\sin^2 x}-\dfrac{e^x\cdot \cos x}{\sin^2 x}
=exsin⁡x−exsin⁡x⋅cos⁡xsin⁡x=\dfrac{e^x}{\sin x}-\dfrac{e^x}{\sin x}\cdot\dfrac{\cos x}{\sin x}
=y−y⋅cot⁡x=y-y\cdot\cot x
=y(1−cot⁡x)=y(1-\cot x)
Grading Criteria

Achievement (u)

  • Correct expression for dydx\dfrac{dy}{dx}.

Merit (r)

  • Correct proof with correct derivative.

Excellence T1

-

Excellence T2

-
Exam Paper (2016)Assessment Schedule (2016)
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