Slope of Ellipse at any Point

Geometric Solution of Slope of Ellipse

Differentiating with respect to the Eccentric Angle

Parametric Equations of an Ellipse
 R = Semi - Major Axis r = Semi - Minor Axis Differentiating with respect to f dy/df = d(r · sinf)/df = r · cosf dx/df = d(R · cosf)/df = R · (– sinf) Since cosf = x/R and sinf = y/r Substituting for sinf and cosf dy/df = r · x/R dx/df = R · (– y/r) Slope at any point on the Ellipse = dy/dx dy/dx = (dy/df) / (dx/df) Substituting for dy/df and dx/df dy/dx = (r · x/R) / (– R · y/r) dy/dx = – (r ² · x) / (R ² · y) Since R = Semi - Major Axis and r = Semi - Minor Axis

 Explicit Differentiation

Resolving the ellipse 4x2 + y2 = 16 in terms of y explicitly as a function of x and differentiating with respect to x.

The relation may be written as two functions:

Differentiating the function in the upper two positive y quadrants:

dy/dx = d(16 - 4x2)1/2/dx

dy/dx = d(16 - 4x2)1/2/d(16 - 4x2) d(16 - 4x2)/dx ... Chain Rule

The solution for the function in the negative y quadrants returns the same terms but with opposite sign.
 Implicit Differentiation

Using Implicit Differentiation to differentiate 4x2 + y2 = 16 with respect to x

d(4x2)/dx + d(y2)/dx = d(16)/dx

d(4x2)/dx + d(y2)/dydy/dx = d(16)/dx ... differentiating implicitly ... Chain Rule

8x + 2ydy/dx = 0

2ydy/dx = -8x

dy/dx = - 4x/y