Question

find the slope of the tangent to the curve at given point Y=xsinx cosx at x=π/2​

Answer 1

Step-by-step explanation:

We have,

$y = x \sin(x) \cos(x)$

$\implies \frac{dy}{dx} = \frac{d}{dx} (x \sin(x) \cos(x) )$

$\implies \frac{dy}{dx} = \sin(x) \cos(x) + x \cos^{2} (x) - x \sin^{2} (x)$

Now, slope of tangent at x = π/2 is

$\implies( \frac{dy}{dx} ) _{x = \frac{\pi}{2} } = \sin( \frac{\pi}{2} ) \cos( \frac{\pi}{2} ) + \frac{\pi}{2} \cos^{2} ( \frac{\pi}{2} ) - \frac{\pi}{2} \sin ^{2} ( \frac{\pi}{2} )$

$\implies( \frac{dy}{dx} ) _{x = \frac{\pi}{2} } = 1 \times 0 + \frac{\pi}{2} \times 0 - \frac{\pi}{2} \times 1$

$\implies( \frac{dy}{dx} ) _{x = \frac{\pi}{2} } = - \frac{\pi}{2}$