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## Implicit differentiation calculator with steps

Implicit differentiation calculator is used to find the differential of implicit function with respect to its variable. This implicit derivative calculator evaluates the implicit equation step-by-step.

The implicit differentiation solver is a type of differential calculator.

## How does implicit differentiation calculator work?

Follow the steps below to solve the problems of implicit function.

• Enter f(x, y) and g(x, y) of the implicit function into the input box.
• Select the variable.
• To write the mathematics keys use the keypad icon next to the input box.
• Click the calculate button to get the dy/dx of the given function.
• Hit the reset button to enter a new input.

## What is implicit differentiation?

Implicit differentiation is a process of finding the differential of a dependent variable in an implicit function by expressing the differential of the dependent variable as a symbol and by differentiating each term separately.

Implicit differentiation calculates the dy/dx of the given equation. This type of differentiation solves the equations without taking y as a constant w.r.t “x”. For example, the implicit differential of $$y^2$$ w.r.t “x” is $$2y\frac{dy}{dx}$$.

The general equation of implicit is:

$$f\left(x,y\right)=g\left(x,y\right)$$

To calculate the implicit differentiation of the equation, we have to apply the differential on both sides of the equation. dy/dx calculator provides the accurate result of the implicit function.

## How to calculate implicit differentiation?

Following are a few examples solved by our implicit differentiation solver.

Example 1

Calculate the implicit differentiation of $$3x^2y+4y^2=23$$ w.r.t “x”.

Solution

Step 1: Use the differentiation notation in the given implicit function.

$$\frac{d}{dx}\left(3x^2y+4y^2\right)=\frac{d}{dx}\left(23\right)$$

Step 2: Differentiate each term separately and apply the power, product, and constant rules.

$$\frac{d}{dx}\left(3x^2y\right)+\frac{d}{dx}\left(4y^2\right)=\frac{d}{dx}\left(23\right)$$

$$y\frac{d}{dx}\left(3x^2\right)+3x^2\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(4y^2\right)=\frac{d}{dx}\left(23\right)$$

$$y\left(3\cdot 2x^{2-1}\right)+3x^2\frac{d}{dx}\left(y\right)+\left(4\cdot 2y^{2-1}\frac{d}{dx}\left(y\right)\right)=0$$

$$6xy+3x^2\frac{dy}{dx}+8y\frac{dy}{dx}=0$$

Step 3: Now separate the dy/dx term.

$$3x^2\frac{dy}{dx}+8y\frac{dy}{dx}=-6xy$$

$$\left(3x^2+8y\right)\frac{dy}{dx}=-6xy$$

$$\frac{dy}{dx}=-\frac{6xy}{\left(3x^2+8y\right)}$$

Example 2

Calculate the implicit differentiation of $$3x^2+6y^2=3x$$ w.r.t “x” by using the chain rule.

Solution

Step 1: Use the differentiation notation in the given implicit function.

$$\frac{d}{dx}\left(3x^2+6y^2\right)=\frac{d}{dx}\left(3x\right)$$

Step 2: Now differentiate the above terms separately and apply the power rule.

$$\frac{d}{dx}\left(3x^2\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)$$

$$3\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)$$

$$3\left(2x\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)$$

$$6x+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)$$

Step 3: Apply the chain rule.

$$\frac{d}{dx}\left(y^2\right)=\frac{du^2}{du}\cdot \frac{du}{dx}where\:u=y\:\&\:\frac{d}{du}\left(u^2\right)=2u$$

Step 4: Simplify the expression.

$$6x+6\left(2y\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(3x\right)$$

$$6x+12\left(\frac{d}{dx}\left(y\right)\right)y=\frac{d}{dx}\left(3x\right)$$

Step 5: Using chain rule

$$\frac{d}{dx}\left(y\right)=\frac{dy\left(u\right)}{dx}\cdot \frac{du}{dx},\:where\:u=x\:\&\:\frac{d}{du}\left(y\left(u\right)\right)=y'\left(u\right)$$

$$6x+\left(\frac{d}{dx}\left(x\right)\right)y'\left(x\right)12y=\frac{d}{dx}\left(3x\right)$$

Step 6: The derivative of 3x is 3.

$$6x+y'\left(x\right)12y=3$$

$$6x+12y\frac{dy}{dx}=3$$

$$12y\frac{dy}{dx}=3-6x$$

$$\frac{dy}{dx}=\frac{3-6x}{12y}$$

## Table of some derivatives of implicit functions

Here are some examples of implicit functions solved by our implicit function calculator.

 Implicit differentiation of Result xy=1 $$\frac{-y}{x}$$ xy+sin(xy)=1 $$\frac{-y}{x}$$ xy-x+2y=1 $$\frac{1-y}{x+2}$$ sqrt(xy)=x^2y+1 $$\frac{4xy\sqrt{xy}-y}{x\:-2x^2\sqrt{xy}}$$ e^xy $$\frac{ye^{xy}}{1-xe^{xy}}$$

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