The following problems require the use of implicit differentiation. Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives. The majority of differentiation problems in first-year calculus involve functions *y* written EXPLICITLY as functions of *x* . For example, if

,

then the derivative of *y* is

.

However, some functions *y* are written IMPLICITLY as functions of *x* . A familiar example of this is the equation

*x*^{2} + *y*^{2} = 25 ,

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 at the point (3, -4) .

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

,

i.e.,

.

Thus, the slope of the line tangent to the graph at the point (3, -4) is

.

Unfortunately, not every equation involving *x* and *y* can be solved explicitly for *y* . For the sake of illustration we will find the derivative of *y* WITHOUT writing *y* explicitly as a function of *x* . Recall that the derivative (D) of a function of *x* squared, (*f*(*x*))^{2} , can be found using the chain rule :

.

Since *y* symbolically represents a function of *x*, the derivative of *y*^{2} can be found in the same fashion :

.

Now begin with

*x*^{2} + *y*^{2} = 25 .

Differentiate both sides of the equation, getting

*D* ( *x*^{2} + *y*^{2} ) = *D* ( 25 ) ,

*D* ( *x*^{2} ) + *D* ( *y*^{2} ) = *D* ( 25 ) ,

and

2*x* + 2 *y y*' = 0 ,

so that

2 *y y*' = - 2*x* ,

and

,

i.e.,

.

Thus, the slope of the line tangent to the graph at the point (3, -4) is

.

This second method illustrates the process of implicit differentiation. It is important to note that the derivative expression for explicit differentiation involves *x* only, while the derivative expression for implicit differentiation may involve BOTH *x* AND *y* .

The following problems range in difficulty from average to challenging.

*PROBLEM 1 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for*x*^{3}+*y*^{3}= 4 .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for (*x*-*y*)^{2}=*x*+*y*- 1 .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for*y*=*x*^{2}*y*^{3}+*x*^{3}*y*^{2}.Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for*e*^{xy}=*e*^{4x}-*e*^{5y}.Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Assume that*y*is a function of*x*. Find*y*' =*dy*/*dx*for .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Find an equation of the line tangent to the graph of (*x*^{2}+*y*^{2})^{3}= 8*x*^{2}*y*^{2}at the point (-1, 1) .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Find an equation of the line tangent to the graph of*x*^{2}+ (*y*-*x*)^{3}= 9 at*x*=1 .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Find the slope and concavity of the graph of*x*^{2}*y*+*y*^{4}= 4 + 2*x*at the point (-1, 1) .Click HERE to see a detailed solution to problem 12.

*PROBLEM 13 :*Consider the equation*x*^{2}+*xy*+*y*^{2}= 1 . Find equations for*y*' and*y*'' in terms of*x*and*y*only.Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Find all points (*x*,*y*) on the graph of*x*^{2/3}+*y*^{2/3}= 8 (See diagram.) where lines tangent to the graph at (*x*,*y*) have slope -1 .Click HERE to see a detailed solution to problem 14.

*PROBLEM 15 :*The graph of*x*^{2}-*xy*+*y*^{2}= 3 is a "tilted" ellipse (See diagram.). Among all points (*x*,*y*) on this graph, find the largest and smallest values of*y*. Among all points (*x*,*y*) on this graph, find the largest and smallest values of*x*.Click HERE to see a detailed solution to problem 15.

*PROBLEM 16 :*Find all points (*x*,*y*) on the graph of (*x*^{2}+*y*^{2})^{2}= 2*x*^{2}-2*y*^{2}(See diagram.) where*y*' = 0.Click HERE to see a detailed solution to problem 16.

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