Homework SourseThe equation sin xy y defines y implicitly as a function of
The equation sin xy = y defines y implicitly as a function of x.
Find the slope y\'(pi/3,1/2) at the point x=pi/3, y=1/2.
Implicitly differentiate the equation with respect to x to find a formula for y \' in terms of x and y.
y \'(x,y) =
Find the slope y\'(pi/3,1/2) at the point x=pi/3, y=1/2.
Implicitly differentiate the equation with respect to x to find a formula for y \' in terms of x and y.
y \'(x,y) =
Solution
Differentiating sin(xy) = y implicitly: y\' = cos(xy)(y + xy\') = ycos(xy) + xy\'cos(xy) y\' - xy\'cos(xy) = ycos(xy) y\'(1 - xcos(xy) = ycos(xy) y\' = (ycos(xy))/(1 - xcos(xy)) y\'(x,y) = (ycos(xy))/(1 - xcos(xy)) <- Here\'s the formula. Now to find the slope at (pi/3, 1/2): y\'(pi/3, 1/2) = ((1/2)cos(pi/6)) / (1 - (pi/3)cos(pi/6) = (1/2)(sqrt(3)/2) / (1 - (pi/3)(sqrt(3)/2)) = (sqrt(3)/4) / ((5 - pi*sqrt(3))/6) = (3sqrt(3)) / (10 - 2pi*sqrt(3))
