Question

F(x)= (5x/sinx) - 3xsinx Solve for dy/dx at x = Pi over 2

Answer 1

`((dy)/(dx))x_{= (\pi )/(2) } = f^(1) ((\pi )/(2) ) =2`

Explanation:

Step(i):-

Given that

`f(x) = (5x)/(sinx) -3xsinx`

Apply (UV )¹ = UV¹+VU¹

`(d)/(dx) ((U)/(V) ) = (V U^(l) -VU^(l) )/(V^(2) )`

Step(ii):-

`f^(1) (x) = 5(sin x(1)-x(cos x))/(sin^(2)x ) - 3(x cos x + sin x(1))`

put
`x = (\pi )/(2)`

`f^(1) ((\pi )/(2) ) = 5(sin (\pi )/(2) (1)-(\pi )/(2) (cos (\pi )/(2) ))/(sin^(2)((\pi )/(2) ) ) - 3((\pi )/(2) cos (\pi )/(2) + sin((\pi )/(2)) (1))`

we know that

`cos((\pi )/(2)) = 0`

`sin((\pi )/(2) ) = 1`

`f^(l) ((\pi )/(2) ) = (5(1-0))/(1) -3(0+1)`

`f^(1) ((\pi )/(2) ) = 5-3 =2`

Final answer:-

`((dy)/(dx))x_{= (\pi )/(2) } = f^(1) ((\pi )/(2) ) =2`