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.

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. - Press
**show more**to view the step-by-step calculations. - Hit the
**reset**button to enter a new input.

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.

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.

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

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

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

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

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 | \(\frac{ye^{xy}}{1-xe^{xy}}\) |

- Implicit differentiation definition & meaning | Merriam-Webster.

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