IMPLICIT DERIVATIVES: HOW THEY ARE SOLVED AND EXERCISES SOLVED - MATHS - 2025

The implicit derivatives are tools used in a differencing technique applied to functions. They are applied when it is not possible, under regular methods, to solve for the dependent variable to be derived. This clearance is carried out as a function of the independent variable.

For example, in the expression 3xy ³ - 2y + xy ² = xy, the expression that defines “y” as a function of “x” cannot be obtained. So that by deriving the differential expression dy / dx can be obtained.

How are implicit derivatives solved?

To solve an implicit derivative, we start with an implicit expression. For example: 3xy ³ - 2y + xy ² - xy = 0. This has already been solved correctly, however doing so is not a necessary condition to obtain the derivative of y with respect to x. Then, each of the elements is derived respecting the chain rule for mixed functions:

3xy ³ is composed of 2 variables, therefore d (3xy ³) will be treated as the derivative of a product of functions.

d (3xy ³) / dx = 3y ³ + 3y ^2. (3x) y '= 3y ³ + 9xy ² y'

Where the element y 'is known as “y prime” and represents dy / dx

-2y It is derived according to the law KU = K.U '

d (-2y) = -2 y '

xy ² supposes another differential composed by a product of functions

d (xy ²) = y ² + 2xy y '

-xy is treated homologously

d (-xy) = -y - x y '

They are substituted in equality, knowing that the derivative of zero is zero.

3y ³ + 9xy ² y '- 2 y' + y ² + 2xy y '- y - x y' = 0

The elements that have the term y 'are grouped on one side of the equality

3y ³ + y ² - y = -9xy ² y '+ 2 y' + x y '

The common factor y 'is extracted from the right side of the equality

3y ³ + y ² - y = y '(-9xy ² + x + 2)

Finally the term that multiplies y 'is cleared. Thus obtaining the expression corresponding to the implicit derivative of y with respect to x.

y '= dy / dx = (3y ³ + y ² - y) / (- 9xy ² + x + 2)

Chain rule

In implicit derivation the chain rule is always respected. All differential expressions will be given as a function of the independent variable X. So every variable θ other than X, must include the term dθ / dx after being derived.

This term will appear only in the first degree or with an exponent equal to 1. This quality makes it completely clear under traditional factoring methods. Thus, it is possible to obtain the expression that defines the differential dθ / dx.

The chain rule shows the progressive nature of the differentiation or derivative process. Where for every compound function f, we have that the differential expression of f will be

Operational order

In each formula or law of derivation that is applied, the order of the variables must be taken into account. The criteria associated with the independent variable are respected, without altering its correlation with the dependent variable.

The relationship of the dependent variable at the time of derivation is taken directly; With the exception that this will be considered as a second function, which is why the chain rule criterion for mixed functions is applied.

This can be developed in expressions with more than 2 variables. Under the same principles, all the differentials referring to the dependent variables will be denoted.

Graphically, the same criterion that defines the derivative is handled. While the derivative is the slope of the tangent line to the curve in the plane, the rest of the differentials belonging to the dependent variables (dy / dx, dz / dx) represent planes tangent to the vector bodies described by the multiple variable functions.

Implicit

A function is said to be implicitly defined if the expression y = f (x) can be represented as a multiple variable function F (x, y) = 0 as long as F is defined in the R ² plane.

3xy ³ - 2y + xy ² = xy can be written in the form 3xy ³ - 2y + xy ² - xy = 0

In view of the impossibility of making the function y = f (x) explicit.

History

The differential calculus began to be named by various mathematical researchers around the seventeenth century. The first time it was mentioned was through the contributions of Newton and Leibniz. Both treated the differential calculus from different points of view, but converging in their results.

While Newton focused on differentiation as a speed or rate of change, Leibniz's approach was more geometric. It can be said that Newton attacked the conjectures left by Apollonius of Perge and Leibniz the geometric ideas of Fermat.

The implicit derivation appears immediately when considering the differential and integral equations. These extended Leibniz's geometric concept to R ³ and even to multidimensional spaces.

Applications

Implicit derivatives are used in various situations. They are common in exchange rate problems between related variables, where, depending on the sense of the study, the variables will be considered dependent or independent.

They also have interesting geometric applications, such as in reflection or shadow problems, on figures whose shape can be mathematically modeled.

They are frequently used in the areas of economics and engineering, as well as in various investigations of natural phenomena and experimental buildings.

Solved exercises

Exercise 1

Define the implicit expression that defines dy / dx

Each element of the expression is differentiated

Establishing the chain rule in each competent case

Grouping on one side of equality the elements that have dy / dx

It is factored using the common factor

It is solved obtaining the expression sought

Exercise 2

Define the implicit expression that defines dy / dx

Expressing the derivatives to be carried out

Deriving implicitly according to chain rule

Factoring common elements

Grouping the term dy / dx on one side of the equality

Common factor to the differential element

We isolate and obtain the expression sought

References

1. Calculus of a Single Variable. Ron Larson, Bruce H. Edwards. Cengage Learning, Nov 10 2008
2. The Implicit Function Theorem: History, Theory, and Applications. Steven G. Krantz, Harold R. Parks. Springer Science & Business Media, Nov 9. 2012
3. Multivariable Analysis. Satish Shirali, Harkrishan Lal Vasudeva. Springer Science & Business Media, Dec 13. 2010
4. System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg. John Wiley & Sons, Mar 7 2012
5. Calculus: Mathematics and Modeling. William Bauldry, Joseph R. Fiedler, Frank R. Giordano, Ed Lodi, Rick Vitray. Addison Wesley Longman, Jan 1 1999