NCEA Calculus 2018 ONE(c - Solution & Explanation

Question

Find the gradient of the curve

y = \ln|\sin^2 x|

at the point where

x = \dfrac{\pi}{6}

You must use calculus and show any derivatives that you need to find when solving this
problem.

Official Answer

y=\ln|\sin^2 x|

\dfrac{dy}{dx}=\dfrac{2\sin x\cos x}{\sin^2 x}

=\dfrac{2\cos x}{\sin x}

OR

y=\ln|\sin^2 x|

=2\ln|\sin x|

\dfrac{dy}{dx}=\dfrac{2\cos x}{\sin x}

etc

When

x=\dfrac{\pi}{6}

,

\dfrac{dy}{dx}=\dfrac{2\cos\dfrac{\pi}{6}}{\sin\dfrac{\pi}{6}}

=2\sqrt{3}

(=3.4641)

Grading Criteria

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Video Explanation

NCEA Level 3 Calculus Differentiation 2018 NZQA Exam - Worked Answers by infinityplusone(starts at 4:12)

2018 ONE(c)