The slope of the tangent line to the ellipse at the point -

Question

The solutions (x,y) of the equation x² + 16y² = 16 form an ellipse as pictured below. Consider the point P as pictured, with x-coordinate 2. 

(a) Let h be a small non-zero number and form the point Q with x-coordinate 2+h, as pictured. The slope of the secant line through PQ, denoted s(h), is given by the formula.

(b) Rationalize the numerator of your formula in (a) to rewrite the expression so that it looks like f(h)/g(h), subject to these two conditions: (1) the numerator f(h)defines a line of slope -1, (2) the function f(h)/g(h) is defined for h=0. When you do this

f(h) = ___

g(h) = ___

(c) The slope of the tangent line to the ellipse at the point P is

lim_h→0 s(h) = ___