Implicit+Differentiation

=**Implicit Differentiation** = **Implicit Differentiation: The process by which we find //dy/dx//.** In mathematics, some equations in //x// and //y// do not explicitly define //y// as a function //x// and cannot be easily manipulated to solve for //y// in terms of //x//, even though such a function may exist. When this occurs, it is implied that there exists a function //y// = //f//( //x//) such that the given equation is satisfied. The technique is known as //**implicit differentiation.**// Implicit differentiation allows us to find the derivative of //y// with respect to //x// without having to solve the given equation for //y//. Always remember that the chain rule must be used whenever the function //y// is being differentiated because of our assumption that //y// may be expressed as a function of //x//.

**Example 1:** Find if //x//2 //y//3 − //xy// = 10. Differentiating implicitly with respect to //x//, you find that
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which represents a circle of radius five centered at the origin. Suppose that we wish to find the slope of the line tangent to the graph of this equation.


How could we find the derivative of //y// in this instance ? One way is to first write //y// explicitly as a function of //x//. Thus, //x//2 + //y//2 = 25 , //y//2 = 25 - //x//2 , and , where the positive square root represents the top semi-circle and the negative square root represents the bottom semi-circle. Since the point (3, -4) lies on the bottom semi-circle given by , the derivative of //y// is , *Be sure to move the 1/2 to the front of the equation, and then subtract one from the power. . Thus, the slope of the line tangent to the graph at the point (3, -4) is

More example problems: **For each of the following equations, find** dy/dxby implicit differentiation. Example 1:

Example 2: Example 3:

Example 4

Work Cited
Bosse, Alexa. "Implicit Differentiation." //Visual Calculus//. 2001. Web. 20 May 2011. [].

Kouba, Duane. "Implicit Differentiation Problems." 23 June 1998. Web. 21 May 2011. []. "Implicit Differentiation." //Cliffnotes//. 2001. Web. 21 May 2011. .