The equation of the tangent line to the curve y = 2x sin x at the point \(\left(\frac \pi 2, \pi\right)\) is

Concept:

The equation of tangent at point (x1 , y1) with slope m is given by 

\(\rm (y - y_1) = m (x - x_1)\)

Calculations:

Given curve is y = 2x sin x 

Taking derivative on both side, we get

\(\rm \dfrac {dy}{dx}= 2x \;cos\;x + 2 \;sin\;x\)

Put x = \(\rm \dfrac{\pi}{2}\) to find the equation of tangent at the point \(\left(\frac \pi 2, \pi\right)\).

\(\rm \dfrac {dy}{dx}= 2 \dfrac {\pi}{2} \;cos\; \dfrac {\pi}{2} + 2 \;sin\; \dfrac {\pi}{2}\)

\(\rm \dfrac {dy}{dx}= 2\)

The equation of tangent at point (x₁ , y₁) with slope m is given by 

\(\rm (y - y_1) = m (x - x_1)\)

\(\rm (y - {\pi}) = 2 (x - \dfrac{\pi}{2})\)

y = 2x

Hence, the equation of the tangent line to the curve y = 2x sin x at the point \(\left(\frac \pi 2, \pi\right)\) is 2x.