# Implicit Differentiation

# Implicit Differentiation

## Definition

**Implicit differentiation**is a technique used to find the derivative of a relation, y as a function of x, where it is difficult to explicitly solve for y.

## Technique

- In order to use implicit differentiation, we differentiate both sides of the equation with respect to x.
- This involves treating y as a function of x, and making use of the
**chain rule**when necessary.

## Example of Applying Implicit Differentiation

- For an equation such as x^2 + y^2 = 1, we don’t have y explicitly solved in terms of x.
- Applying implicit differentiation, the derivative is produced as (2x + 2y*dy/dx) = 0, and then we solve for dy/dx.

## Differentiating with Respect to Other Variables

- Implicit differentiation can also be applied when differentiating with respect to variables other than x.
- In such cases, the chain rule is applied to the other variables.

## Common Mistakes in Implicit Differentiation

- Do not forget to apply the chain rule when differentiating y in respect with x. If you do so, the derivative will be incomplete.
- Do not forget that the derivative of a constant is zero.

## Applications of Implicit Differentiation

- Implicit differentiation plays a key role when working with
**related rates problems**in calculus. - In geometry, it is commonly used when determining the equation of tangents and normals to curves defined by an implicit relation.
- It serves as a fundamental cornerstone in understanding the behaviour of implicitly defined functions or relations.