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Question
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Solution
\[ y = \left( \sin 2x + \cot x + 2 \right)^2 \]
\[ \Rightarrow \frac{dy}{dx} = 2 \left( \sin 2x + \cot x + 2 \right) \left( 2\cos 2x - \ cose c^2 x \right)\]
\[\text { Now,} \]
\[\text { Slope of the tangent }= \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{2}} \]
\[=2\left[ \sin 2\left( \frac{\pi}{2} \right) + \cot \left( \frac{\pi}{2} \right) + 2 \right] \left[ 2\cos 2\left( \frac{\pi}{2} \right) - {cosec}^2 \left( \frac{\pi}{2} \right) \right]\]
\[ = 2 \left( 0 + 0 + 2 \right) \left( - 2 - 1 \right)\]
\[ = - 12\]
\[\text { Slope of the normal }=\frac{- 1}{\left( \frac{dy}{dx} \right)_{x = \frac{\pi}{2}}}=\frac{- 1}{- 12}=\frac{1}{12}\]
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