Calculus
posted by Tyler on .
Find the equation of the tangent line to the curve (piriform)
y^2=x^3(4−x)
at the point (2,16−−ã).
a. Find dy/dx at x=2.
dy/dx=
b. Write the equation of the tangent line to the curve.

y^2 = 4x^3  x^4
2yy' = 12x^2  4x^3
y' = (6x^2  2x^3)/y
y(2) = 4
y'(2) = (6*42*8)/4 = 2
So, now you have a point and a slope. The tangent line at (2,4) is
y4 = 2(x2) 
2y dy/dx = x^3 (1) + 3x^2 (4x)
dy/dx = (x^3 + 12x^2  3x^3)/(2y)
= (6x^2  2x^3)/y
I can't make out your point (2, 16−−ã)
but it should be easy for you
just plug in x=2 and y = whatever into the dy/dx
and that becomes your slope
then use the grade 9 way you learned to find the equation of a line with a given slope and a given point.