help! find dy/dx and the slope of the graph of the polar curve at the given value of w (angle)?

Question

r = f(w) = 6 / (2sinw - 3cosw)
,
w = pi

i need a step by step guide please...

thanks!!!

Answer 1

i assume you mean dr/dw.
We will just use repeated chain rule to differentiate
if we say r=6/u where u = 2sin(w) - 3cos(w)
then dr/dw = (dr/du)*(du/dw)
dr/du = -6/(u^2)
du/dw=2cos(w) + 3sin(w)
dr/dw = -6*(2cos(w) + 3sin(w)) / (2sin(w) - 3cos(w))^2
at w=pi we get
r'(pi) = 12/9=4/3

Answer 2

step by step
x= r cos w
y = r sinw
so dx/dw = dr/dw*cosw-rsin w
dy/dw = dr/dw sin w +r cos w
so
dy/dx= (dr/dw sin w+r cos w)/(dr/dw*cos w-r sin w)

in your case dr/dw = 6(-2 cos w -3sin w)/(2sin w -3 cos w) ^2
Evaluate all at w=pi
dr/dw= 6(2)/9 = 4/3 f(pi)=2
so dy/dx= (4/3*0-2)/(-4/3)=6/4=3/2