Implicit Differentiation Calculator with steps | Find dy/dx or dx/dy

Implicit Differentiation Calculator

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Sometimes we need to calculate the derivative of implicit functions, in which one variable can not be explicitly expressed in terms of the other. To help you solve this type of derivatives, we put in your hands the Implicit Differentiation Calculator with Steps.

To calculate the derivative using implicit differentiation calculator you must follow these steps:

  1. Enter the implicit function in the calculator, for this you have two fields separated by the equals sign. The functions must be expressed using the variables x and y.
  2. Select dy/dx or dx/dy depending on the derivative you need to calculate.
  3. Press the “Calculate” button to get the detailed step-by-step solution.

What is an implicit function?

An implicit function is that function in which the dependent variable y is not explicitly isolated. An implicit function is expressed by an equality of the form: f(x,y)=0.

Let’s look at some examples of implicit functions:

  • y = x2y + ex
  • xy = 3x2 – x + 2

  • 7x – y + 15 = 0

In orther words, an implicitly defined function is a function that is not given in the form of an equation with an independent variable and a dependent variable, but rather as an equation that relates the independent and dependent variables in some other way.

For example, consider the equation x2 + y2 = 1. This equation defines a circle with radius 1 centered at the origin. The function y is not explicitly given in terms of x, but rather it is implicitly defined as a function of x by the equation.

What is Implicit Differentiation?

Implicit differentiation is the technique that allows us to obtain the derivative of the implicit function. The implicit differentiation technique is based on the use of the chain rule to find dy/dx or dx/dy.

In the next section we will explain step by step how to do the implicit differentiation.

How to do Implicit Differentiation

Because the implicit functions have the form f(x,y)=0, we cannot start the differentiation process directly, since the dependent and independent variables are intermeshed. Here are the steps you need to follow to do implicit differentiation:

We will assume that we want to find dy/dx by implicit differentiation.

  1. Differentiate each term with respect to the independent variable on both sides of the equals sign. Note that y is a function of x. Consequently, for example, d/dx (sin(y)) = cos(y)⋅dy/dx due to the use of the chain rule.
  2. Rewrite the equation so that all terms containing dy/dx are on the left and all terms not containing dy/dx are on the right.
  3. Isolates dy/dx.

To better understand how to do implicit differentiation, we recommend you study the following examples.

Examples

Example 01: From the equation x2 + y2 = 25, find dy/dx by implicit differentiation

  1. Differentiate both sides of the equation:

Example 01 - find dy dx by implicit differentiation - step 01

  1. Keep the terms with dy/dx on the left. Move the remaining terms to the right:

Example 01 - find dy dx by implicit differentiation - step 2

  1. Divide both sides of the equation by 2y:

Example 01 - find dy dx by implicit differentiation - step 3

Example 02: Using implicit differentiation to find dy/dx of this function: cos(y + 1) + xy = xy3

  1. Differentiate each side of the equation with respect to x:

  1. Now move all terms with dy/dx to the left side of the equation and all terms without dy/dx move to the right side.

  1. Factor out the dy/dx from the left side terms.

  1. Isolate dy/dx to get the solution.

Applications

Some of the main applications of implicit differentiation are listed below:

  1. Implicit differentiation is used to find the derivative of inverse functions.
  2. Use implicit differentiation to find the equation of the tangent line.
  3. Indifference curves in economics

Implicit Differentiation worksheet

It is important to practice the newly learned concepts in order to understand them perfectly. To do this, we share three worksheets with implicit differentiation practice problems.

Remember that if you need help to solve some of these problems you can use the Implicit Differentiation Calculator to see step by step how to solve them.

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