Math

Implicit derivatives: Formula, Examples And How to Solve?

What are implicit derivatives?

Implicit derivatives are tools used in a differentiation technique applied to functions. They are applied when it is not possible, under regular methods, to isolate the dependent variable that we want to derive. This clearance is done based on the independent variable.

For example, in the expression 3xy 3 – 2y + xy 2 = xy, you cannot get the expression that defines “y” as a function of “x”. So that when differentiating the differential expression dy/dx can be obtained.

How are implicit derivatives solved?

To solve an implicit derivative, we start from 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 are 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(3xy3)/dx = 3y3 + 3y2.(3x) y’ = 3y3 + 9xyy’

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) = -2y’

xy 2 supposes another differential composed of a product of functions

d(xy2) = y2 + 2xy y’

-xy is a homologous form

d(-xy) = -y – x y’

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

3y3 + 9xyy’ – 2 y’ + y2 + 2xy y’ – y – x y’ = 0

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

3y+ y2 – y = -9xyy’ + 2 y’ + x y’

The common factor y’ is extracted in the right member of the equality

3y+ y2 – y = y’ (-9xy2 + x + 2)

Finally, the term that multiplies y’ is solved. Thus obtaining the expression corresponding to the implicit derivative of y with respect to x.

y’ = dy/dx = (3y+ y2 – y)/(-9xy2 + x + 2)

Chain rule in implicit derivatives

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 only appear in the first degree or with an exponent equal to 1. This quality makes it completely clearable under traditional factorization methods. So it becomes 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 composite function f [ g(x) ], we have that the differential expression of f will be

Operational order in implicit derivatives

In each formula or derivation law that is applied, the order of the variables relative to each other 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 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 differentials referring to the dependent variables will be denoted.

Graphically, the same criterion that defines the derivative is used. 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 tangent planes to the vector bodies described by the multiple variable functions.

Implicit  of a function

A function is said to be implicitly defined if the expression y = f(x) can be represented as a multivariable 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 of implicit derivatives

Differential calculus began to be mentioned by various mathematical researchers around the 17th century. The first time it was mentioned was through the contributions of Isaac Newton and Gottfried Leibniz. Both treated differential calculus from different points of view, but converging in their results.

While Newton focused on differentiation as a velocity 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 attacked the geometric ideas of Pierre de Fermat.

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

Applications of implicit derivatives

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

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

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 the equality the elements that have dy/dx

It is factored using the common factor

It is resolved by obtaining the desired expression

Exercise 2

Define the implicit expression that defines dy/dx

Expressing the derivatives to be made

Implicitly differentiating according to the chain rule

Factoring common elements

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

Common factor to the differential element

We solve and obtain the desired expression