Lecture ¹4

# Lecture ¹4. The existence and the differentiability of the implicit function. The implicit functions defined in system of functional equations

Exercises for Lecture #4

The purpose of work:

The differentiation of the implicit function.

Theoretical information and examples

Content

1.9 Differentiation of Implicit Functions

1.9.1 Case of one independent variable

1.9.2 Case of several independent variables

1.9.3 A system of implicit functions

1.9.4 Parametric representation of a function

1.9 Differentiation of Implicit Functions

1.9.1 Case of one independent variable. If the equation f(x, y)= 0, where f(x, y) is a differentiable function of the variables x and y, defines y as a function of x, then the derivative of this implicitly defined function, provided that f'y(x, y) ¹ 0, is given by

Higher order derivatives are found by successive differentiation of (1).

Example 1. Find dy/dx and d²/dy² if

Solution. Denoting the left hand side of this equation by f(x, y), we find the partial derivatives

whence, applying (1), we get

In order to find the second derivative, differentiate with respect to x the first derivative which we have found, taking into consideration the fact that y is a function of x:

1.9.2 Case of several independent variables. Similarly, if the equation F(x,y,z)=0, where F(x, y, z) is a differentiable function of the variables x, y, z, defines z as a function of the independent variables x and y and F'z(x, y, z)¹0, then the partial derivatives of this implicitly represented function can, generally speaking, be found from

Here follows another way of finding the derivatives of the function z: Differentiating the equation F(x, y, z) = 0, we find

whence it is possible to determine dz and therefore

Example 2. Find if

Solution/First method. Denoting the left hand side of this equation by F(x,y,z), we find the partial derivatives

Applying (2), we get

Second method. Differentiating the given equation, we obtain

Hence, we determine dz, i.e., the total differential of the implicit function:

Comparing this result with the formula

we see that

1.9.3 A system of implicit functions. If a system of two equations

defines u and v as functions of the variables x and y and the Jacobian

then the differentials of these functions (and hence also their partial derivatives) may be found from the set of equations

(3)

Example 3. The equations

define u and v as functions of x and y; find

Solution/First method. Differentiating both equations with respect to x, we obtain

whence

Similarly, we find

Second method. By differentiation, we find two equations which connect the differentials of all four variables:

Solving this system for the differentials du and dv, we obtain

whence

1.9.4 Parametric representation of a function. If a function z of the variables x and y is represented parametrically by the equations

x = x(u, v), y = y(u, v), z = z(u, v)

and

then the differential of this function may be found from the system of equations

Knowing the differential dz = pdx + qdy, we find the partial derivatives

Example 4. The function z of the arguments x and y is defined by the equations

Find .

Solution/First method. By differentiation, we find the three equations connecting the differentials of all five variables:

We determine from the first two equations du and dv:

Substituting into the third equation the values of du and dv just found, we have

Hence,

Second method. From the third equation, we can find

Differentiate the first two equations first with respect to x and then with respect to y:

From the first system, we find

from the second system,

Substituting these expressions into (5), we obtain

Control questions

1. What formulae is used to found the derivative of implicit function?