Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Differentiation,2,Implicit Differentiation,2.5,Implicit and Explicit Functions,Implicit and Explicit Functions,Most functions have been expressed in,explicit form.,For example,in the equation,the variable,y,is explicitly written as a function of,x,.,Some functions,however,are only implied by an equation.For instance,the function,y=,1/,x,is defined,implicitly,by the equation,xy=,1.,Explicit form,Suppose you were asked to find,dy,/,dx,for this equation.You could begin by writing,y,explicitly as a function of,x,and then differentiating.,This strategy works whenever you can solve for the function explicitly.,You cannot,however,use this procedure when you are unable to solve for,y,as a function of,x.,Implicit and Explicit Functions,Suppose you were asked to find,dy,/,dx,for this equation.You could begin by writing,y,explicitly as a function of,x,and then differentiating.,This strategy works whenever you can solve for the function explicitly.,You cannot,however,use this procedure when you are unable to solve for,y,as a function of,x.,Implicit and Explicit Functions,Suppose you were asked to find,dy,/,dx,for this equation.You could begin by writing,y,explicitly as a function of,x,and then differentiating.,This strategy works whenever you can solve for the function explicitly.,You cannot,however,use this procedure when you are unable to solve for,y,as a function of,x.,Implicit and Explicit Functions,Suppose you were asked to find,dy,/,dx,for this equation.You could begin by writing,y,explicitly as a function of,x,and then differentiating.,This strategy works whenever you can solve for the function explicitly.,You cannot,however,use this procedure when you are unable to solve for,y,as a function of,x.,Implicit and Explicit Functions,For instance,how would you find,dy,/,dx,for the equation,where it is very difficult to express,y,as a function of,x,explicitly?To do this,you can use,implicit differentiation.,Implicit and Explicit Functions,To understand how to find,dy,/,dx,implicitly,you must realize that the differentiation is taking place,with respect to x.,This means that when you differentiate terms involving,x,alone,you can differentiate as usual.,However,when you differentiate terms involving,y,you must apply the Chain Rule,because you are assuming that,y,is defined implicitly as a differentiable function of,x.,Implicit and Explicit Functions,Example 1,Differentiating with Respect to x,Example 1,Differentiating with Respect to x,Example 1,Differentiating with Respect to x,contd,Example 1,Differentiating with Respect to x,contd,Example 1,Differentiating with Respect to x,contd,Example 1,Differentiating with Respect to x,contd,Example 1,Differentiating with Respect to x,Implicit Differentiation,Implicit Differentiation,Example 2,Implicit Differentiation,Find,dy,/,dx,given that,y,3,+,y,2,5,y,x,2,=4.,Solution:,1.,Differentiate both sides of the equation with respect to,x.,2.,Collect the,dy,/,dx,terms on the left side of the equation and move all other terms to the right side of the equation.,3.,Factor,dy,/,dx,out of the left side of the equation.,4.,Solve for,dy,/,dx,by dividing by(3,y,2+2,y,5).,contd,Example 2,Solution,Example 3,Differentiating with Respect to x,Find tangent line to the graph given by the below through the point:,x,2,(x,2,+x,2,y,2,)=y,2,To see how you can use an,implicit derivative,consider,the graph shown:,From the graph,you can see that,y,is not a function of,x.,Even so,the derivative found in Example 2,gives a formula for the slope of the,tangent line at a point on this graph.,The slopes at several points on the,graph are shown below the graph.,Figure 2.27,Implicit Differentiation,Example 5,Finding the Slope of a Graph Implicitly,Determine the slope of the graph of 3(,x,2,+,y,2,),2,=100,xy,at the point(3,1).,Solution:,Example 5,Solution,contd,At the point(3,1),the slope of the graph is as shown.,This graph is called a,lemniscate.,Example 5,Solution,contd,Board Problems:,