Question

Find the equation of the tangent line to the curve at the given point?

Answer 1

Answer:

The equation of the tangent line

$y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )$

Step-by-step explanation:

Step(i):-

Given function y = x cot (x) ....(i)

Differentiating equation (i) with respective to 'x' , we get

$\frac{dy}{dx} = x (-Co sec^{2} (x)) +cot(x) (1)$

Step(ii):-

The slope of the tangent line

$\frac{d y}{d x} = -x Co-sec^{2} (x) +cot x$

$(\frac{d y}{d x} )x_{=\frac{\pi }{4} } = -\frac{\pi }{4} Co-sec^{2} (\frac{\pi }{4} ) +cot \frac{\pi }{4}$

We will use trigonometry formulas

$Cosec(\frac{\pi }{4} ) = \sqrt{2}$

$sec(\frac{\pi }{4} ) = \sqrt{2}$

$Cot(\frac{\pi }{4} ) = 1$

Now the slope of the tangent

$\frac{dy}{dx} =-\frac{\pi }{4} (\sqrt{2})^{2} )+1$

$\frac{dy}{dx} =-\frac{\pi }{2} +1$

Step(iii):-

Given

Substitute $x=\frac{\pi }{4}$ in y = x cot (x)

               $y = \frac{\pi }{4} cot (\frac{\pi }{4} )$

              $y = \frac{\pi }{4}$

The point of the tangent line  $(x ,y ) = (\frac{\pi }{4} , \frac{\pi }{4} )$

The equation of the tangent line

$y - y_{1} = m ( x - x_{1} )$

$y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )$