# Implicit Differentiation Calculator with Steps

## Calculate implicit derivatives step by step

The implicit differentiation calculator will find the first and second derivatives of an implicit function treating either $y$ as a function of $x$ or $x$ as a function of $y$, with steps shown.

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Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Implicit Differentiation Calculator is a tool that helps you find the derivative of an implicit function quickly and correctly. Our calculator, which has an intuitive interface, implements the implicit differentiation method and explains each step of the process.

## How to Use the Implicit Differentiation Calculator?

• ### Input

Enter the function you want to differentiate in the appropriate field. Ensure your function includes both $x$ and $y$ variables. Choose with respect to which variable to differentiate. Optionally, enter the point at which the derivative should be evaluated.

• ### Calculation

Once you've inputted all the data, click the "Calculate" button.

• ### Result

Within moments, the calculator will display the derivative. If the point is entered, the value of the derivative at that point will be shown.

## What Is Implicit Differentiation?

Implicit differentiation is an important technique in calculus. It is used for functions where one variable (typically $y$) is not explicitly expressed in terms of another variable. Instead of initially isolating $y$ and proceeding to differentiate, this approach allows us to find the derivative immediately.

The Core Principle of Implicit Differentiation

To understand the process of differentiating an implicit function, it is necessary to understand that each term is explicitly or implicitly dependent on the independent variable. This allows us to use the chain rule.

Suppose we are given a general implicit function $F(x,y)=0$. Differentiating it with respect to x, you get the following:

$$\frac{d}{dx}F(x,y)=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}\frac{dy}{dx}=0$$

Now, rearrange and solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$$

This is the derivative of $y$ with respect to $x$.

Example

Consider the following equation of a circle:

$$x^2+y^2=25$$

In differentiating both sides with respect to $x$, we get that

$$2x+2y\frac{dy}{dx}=0$$

From this, we can rearrange and solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=-\frac{x}{y}$$

This result gives us the slope of the tangent line to the circle at any point $(x,y)$.

## What Are the Benefits of Using Implicit Differentiation?

Implicit differentiation offers many advantages, mainly when dealing with complex equations and functions. Here are some benefits:
• Handling Complex Functions: Implicit differentiation is invaluable when dealing with functions that are difficult or impossible to rewrite as explicit functions of $y$ in terms of $x$.
• Direct Approach: This method skips the step of isolating $y$ and goes straight to finding the derivative, saving time and potential errors from added manipulations.
• Broader Scope: In disciplines such as physics and engineering, functions are often given in an implicit form rather than in an explicit one. Implicit differentiation handles such situations.
• Analyzing Interdependent Variables: Not all relationships between variables are simple. Implicit differentiation allows us to understand the rate of change of variables that are intricately related.
• Broad Applicability: Implicit differentiation is critical for studying the behavior of curves in coordinate geometry, such as understanding the slopes of tangents to ellipses, hyperbolas, or other conic sections.

## Why Choose Our Implicit Differentiation Calculator?

• ### Accuracy

Our calculator is built to give you accurate results. It eliminates any difficulties associated with implicit differentiation.

• ### Versatility

The calculator can handle a wide range of implicit functions, from basic to complex.

• ### Ease of Use

With its intuitive design, even those new to calculus can effortlessly navigate and understand the process.

• ### Fast Processing

Get your results instantly, without delays and long waiting.

### FAQ

#### What is Implicit Differentiation?

Implicit differentiation is a technique used in calculus to determine the derivative of functions that aren't explicitly written in terms of one variable. For instance, when an equation involves both $x$ and $y$ without $y$ being isolated, implicit differentiation allows to find $\frac{dy}{dx}$ directly without first solving for $y$.

#### Is implicit differentiation the same as partial differentiation?

No, they are not the same. Implicit differentiation is used when dealing with equations where a variable (commonly $y$) isn't isolated. On the other hand, partial differentiation is a technique used in multivariable calculus where the derivative of a function with respect to one variable is taken, treating all other variables as constants.

#### Where is implicit differentiation used?

Implicit differentiation is widely used in calculus, especially when dealing with complex equations where one variable is not explicitly expressed in terms of the other. It's useful in areas like physics, engineering, and economics, where relationships between variables can only sometimes be expressed explicitly. This method simplifies the differentiation process, making it easier to find rates of change or slopes in such situations.

#### What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of functions that are not explicitly defined in terms of one variable. Instead of isolating a specific variable (like $y$) and differentiating it, this approach enables direct differentiation of the implicit function, resulting in a derivative that involves both variables.