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: For the equation , differentiating both sides with respect to x gives:

Solving for :

This formula for the slope applies at any point on the circle, not just below the x-axis.

Application: Finding Tangent Lines

To find the slope of the tangent line to a curve defined implicitly, use the derivative obtained from implicit differentiation. For example, at the point (3, -4) on the circle :

• Slope:

• At (3, -4):

Note: The formula is valid for any point on the circle, not just those below the x-axis.

Techniques of Differentiation

Implicit Differentiation with Trigonometric and Product Terms

When the equation involves more complex terms, such as products or trigonometric functions, apply the product rule and chain rule as needed. Consider the equation :

• Differentiating both sides with respect to x:

Applying the chain and product rules:

Rearrange to solve for :

• Example: The graph of is shown in the figure below.

Summary Table: Steps for Implicit Differentiation

Additional info: Implicit differentiation is essential for finding derivatives of curves not easily written as , such as circles, ellipses, and more complex relations involving trigonometric or exponential terms.