Implicit Differentiation Calculator

Implicit Differentiation Calculator

Implicit differentiation calculator is used to find the derivative of a dependent variable in an implicit function. It differentiates each term separately. It provides explanations for every step and differentiation rules used for implicit differentiation.

The derivative will be shown in three alternative forms. You can choose to differentiate with respect to x or y.

What is implicit differentiation?

Implicit differentiation is used to find the derivative of an equation. It is the method of performing differentiation on both sides with respect to a variable.

Implicit equations cannot be expressed in terms of one variable. E.g: 

3x² - 5y² + 9x = 25 - 15y

Normally, an equation is first adjusted so that it is expressed as a function to find derivatives. But when it comes to implicit equations, this is not possible. This is why there is a separate calculator to perform differentiation implicitly. 

How to perform implicit differentiation?

Differentiation is a long and difficult process. And implicit differentiation is even more tiresome. This is why the best option is the implicit derivative calculator above to find derivatives with steps.

But you should understand the manual process as well. This problem can be solved with the help of an example.

Example:

Find the derivative of x² + y² = 5 with respect to x.

Solution:

It is important to choose a differentiation notation. The above calculator uses dy/dx. We will also use the same notation in this example.

$$\frac{d}{dx}\left(x^2\:+\:y^2\right)=\frac{d}{dx}\left(5\right)$$

Step 1: Differentiate the expression using linearity. i.e., differentaite each term separately.

$$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(y^2\right)=\frac{d}{dx}\left(5\right)$$

Step 2: Use the power rule.

$$\frac{d}{dx}\left(y^2\right)+2x=\frac{d}{dx}\left(5\right)$$

 Step 3: Use the chain rule, where u = y and $$\frac{d}{du}\left(u^2\right)=\:2u$$ 

$$2x+2y\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(5\right)$$

Step 4: Using the chain rule, where u = x and $$\frac{d}{du}\left(y\left(u\right)\right)=\:y'\left(u\right)$$

$$dx+\left(\frac{d}{dx}\left(x\right)\right)y'\left(x\right)2y=\frac{d}{dx}\left(5\right)$$

The derivative of x is 1.

$$\left(2x+1\right)\:2y\:y'\left(x\right)=\frac{d}{dx}\left(5\right)$$

Step 5: Use the coefficient rule.

$$2x+2y\:y'\left(x\right)=0$$

$$2y\:y'\left(x\right)=-2x$$

Divide both sides by 2.

$$y'\left(x\right)-\frac{x}{y}$$

To revert this differentiaion, you can use an antiderivative calculator.