Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2

The equation of the tangent line is    ======    the derivative at a point is the slope of the tangent line to that point.to implicitly differentiate, we assume that  is a function of  (so we need to apply chain rule when differentiating y terms)differentiate both sides    for the left-hand side, use sum rule for differentiation: differentiate each term    for the right-hand side, we need to use the chain rule.i will deal with right-hand side separately here, and then add it back to original equation after    let's just connect this together now    we do not want to simplify this. you can if you want, but it involves  a lot of  time and unnecessary algebra. you might make mistakes. i might go over the character limit for these answers (around 5000 characters). stuff like that. we do not need to solve for  in terms of  and  when we want to find the derivative at a point for an implicit curve.since we are interested in the point  , substitute in 0 for all  and 0.5 for all  . then solve for  , which is the slope of the tangent line.that x-value of 0 will really us out.    since  is 1 at the point , the slope of the tangent line at point  is 1.therefore, the slope of the line using point-slope form is   

Idon’t know what the answer is i wish i could

Period is the time it takes for one “cycle” to complete. in the parent sine function, period is 2*pi since sin(0) = 0, sin(pi/2) = 1, sin(pi) = 0, sin(3pi/2) = -1, and sin(2pi) = 0. at that point, the sequence repeats itself. i’m not sure if your graph is similar to the sine function since you didn’t provide a picture, but you can find the period by looking at a graph.