Techniques of Differentiation

Implicit Differentiation

Implicit differentiation is a method used to find the derivative of functions that are not explicitly solved for one variable in terms of another. This technique is especially useful when dealing with equations where y is defined implicitly as a function of x, rather than explicitly.

• Implicit Function: An equation involving both x and y that defines y as a function of x indirectly.

• Key Idea: Differentiate both sides of the equation with respect to x, treating y as a function of x (i.e., apply the chain rule when differentiating terms involving y).

Example: Implicit Differentiation of a Circle

Consider the equation of a circle: . To find , differentiate both sides with respect to x:

• Solving for gives:

This formula applies to any point on the circle, not just those below the x-axis. Note that the derivative involves both variables x and y.

Example: Implicit Differentiation with Trigonometric Terms

Given , find .

• Differentiating both sides with respect to x:

• Apply the product rule to :

• Substitute back:

• Expand and collect terms involving :

• Factor :

• Finally, solve for :

Key Steps in Implicit Differentiation

• Step 1: Differentiate both sides of the equation with respect to x.

• Step 2: Apply the chain rule to terms involving y (i.e., ).

• Step 3: Collect all terms involving on one side of the equation.

• Step 4: Factor and solve for .

Applications and Observations

• Implicit differentiation is essential for finding slopes of curves not given as explicit functions.

• It is widely used in calculus for analyzing curves, especially those defined by equations involving both x and y.

• Historical note: The folium of Descartes () is a classic example of a curve studied using implicit differentiation.