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.