Implicit Differentiation Calculator

Implicit differentiation Calculator

To find the derivatives, input the function and choose a variable from this implicit differentiation calculator. After that hit ‘calculate’.

The implicit derivative calculator performs a differentiation process on both sides of an equation. This differentiation (dy/dx) calculator provides three alternative answers.

You have the choice of variable in this calculator. Users can see a detailed explanation of the differentiation process in the results. 

What is Implicit differentiation?

Implicit means “unexpressed”. An implicit equation cannot be expressed in terms of any variable. It is always vague. So the normal differentiation process is not applicable to such equations. 

An example of such an equation is 3xy + 4x² = 6y + 25. You can differentiate this equation only implicitly.

In implicit differentiation, we find the derivative of the equations with respect to one variable. The derivative found by different variables is different.

Implicit differentiation example:

In implicit differentiation, the chain rule is used almost every time. It is difficult to apply and the possibility of error is great. To escape any problem, use the implicit differentiation calculator with steps for free.

Example:

How to perform implicit differentiation on x² + 4y² = 1 with respect to x.

Solution:

Apply differentiation notation on both sides.

$$ \frac{d}{d x}(x^2 = 4y^2 )=\frac{d}{d x}(1) $$

1. Differentiate the sum term by term and factor out constants.

$$\left(\frac{d}{d x}\left(x^{2}\right)-4\left(\frac{d}{d x}\left(y^{2}\right)\right)\right)=\frac{d}{d x}(1)$$

2. Use the power rule.

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

5. Using the chain rule \( \small\frac{d}{d x}(y^2)=\frac{du^2}{d u} \space \frac{du}{d x} \text{where u = y and } \frac{d}{du}(u^2) = 2u \)

$$ 2 x-4\left[2 y\left(\frac{d}{d x}(y)\right)\right]=\frac{d}{d x}(1) $$

4. Simplify the expression.

$$ 2 x-8\left(\frac{d}{d x}(y)\right)y=\frac{d}{d x}(1) $$

5. Using the chain rule \( \small\frac{d}{d x}(y)=\frac{dy(u)}{d u} \space \frac{du}{d x} \text{where u = x and } \frac{d}{du}(y(u)) = y'(u) \)

$$ 2 x \left(\frac{d}{d x}(y)\right)y'(x)8y=\frac{d}{d x}(1) $$

6. The derivative of x is 1.

$$ 2 x-18 y y'(x) = \frac{d}{d x}(1) $$

7. The derivative of the constant term is zero.

$$ 2 x-18 y y'(x) = 0 $$

$$ -18 y y'(x) = -2x $$

$$  y'(x) = \frac{x}{4y} $$