Chapter 2: Differentiation
Section 2.5: Implicit Differentiation
The function f whose rule is given by fx=x2+x+1, is said to be defined explicitly. The function yx whose rule must be extracted from an equation of the form Fx,y=0 is said to be defined implicitly.
A simple example is the circle, defined by x2+y2=9, where y±x= ±9−x2 are two different explicit functions that can be extracted from the equation of the circle. The semicircle above the x-axis is defined by y+x=9−x2; and below, by y−x= −9−x2.
Implicit differentiation is a technique by which y′x can be obtained without necessarily having to solve for yx explicitly. It is merely the Chain rule applied to the identity Fx,yx=0.
If an equation of the form Fx,y=0 defines a function yx implicitly, then Fx,yx≡0, that is, substitution of the expression yx back into the equation results in an expression that is identically zero. This being the case, imagine that yx has been found explicitly, so that Fx,yx is identically zero.
Differentiate Fx,yx with respect to x, and solve for y′x in the resulting equation. Examples 2.5.1-3 illustrate exactly what this process entails.
Extract y+x=9−x2 from x2+y2=9, and show that Fx,y+x=0 is an identity. Obtain y+/x from this explicit representation.
Obtain the derivative of y+x from x2+y2=9, the implicit representation of the circle.
The curve y=yx defined implicitly by the equation x3+y3=3 a x y is called the Folium of Descartes.
Obtain a graph of this curve. In particular, explore the effect of a on the curve.
Obtain y′x by implicit differentiation.
Find all points on this curve where its tangent line is horizontal.
Find all points on this curve where its tangent line is vertical.
Working numerically in the case a=3, find all points on the curve where its slope is 1. Obtain and graph the corresponding tangent lines.
Solve for yx explicitly in the case a=3. Graph the resulting branches.
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