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 
Resolving the ellipse
4x^{2} + y^{2} = 16
in terms of y explicitly as a function of x and differentiating with respect to x.
Differentiating the function in the upper two positive y quadrants: dy/dx = d(16  4x^{2})^{1/2}/dx dy/dx = d(16  4x^{2})^{1/2}/d(16  4x^{2}) • d(16  4x^{2})/dx ... Chain Rule The solution for the function in the negative y quadrants returns the same terms but with opposite sign.
Using Implicit Differentiation to differentiate
4x^{2} + y^{2} = 16
with respect to x
