Implicit Differentiation

Definition and Motivation

In calculus, implicit differentiation is a technique used to find the derivative dy/dx when a function is defined by an equation involving both x and y, rather than by an explicit formula of the form y = f(x). This method is especially useful when it is difficult or impossible to solve the equation explicitly for y.

• Key Principle: Differentiate both sides of the equation with respect to x, treating y as a function of x.

• Chain Rule: When differentiating terms involving y, multiply by dy/dx.

• Applications: Used to compute slopes and tangent lines for curves defined implicitly, such as circles and more complex curves.

Example 1: Circle

Given the equation of a circle:

• Differentiate both sides:

• Result:

• Solve for :

Equation of the tangent line at (3,4):

• Slope:

• Point-slope form:

Example 2: Implicit Curve

Given

• Differentiate:

• Group terms:

• Factor:

• Solve:

Tangent at (3,3): Equation:

Horizontal Tangent: Set Substitute into the original equation and solve for and .

Example 3: Trigonometric Implicit Equation

Given

• Differentiate both sides:

  • Left:

  • Right:

• Collect terms and solve:

Second Derivative by Implicit Differentiation

Given

• First derivative:

• Second derivative: Use the quotient rule and substitute :

Substitute and simplify: Since ,

Derivatives of Inverse Trigonometric Functions

General Formulas

Derivation Example:

• Let

• Differentiating both sides:

• Since , (principal value)

• Thus,

Derivation Example:

• Let

• Differentiating:

• Since ,

Derivation Example:

• Let

• Differentiating:

• Since ,

Examples Involving Inverse Trigonometric Functions

• Differentiate :

  • Rewrite:

  • Apply Chain Rule:

  • Final answer:

• Differentiate :

  • Product Rule:

  • Chain Rule:

  • Final answer:

Further Examples of Implicit Differentiation

• Given :

  • Differentiate:

  • Group :

  • Solve:

• Given :

  • Differentiate both sides and collect terms:

  • Final answer:

• Given :

  • Differentiate:

  • Solve:

• Given :

  • Differentiate:

  • Solve:

• Given :

  • Differentiate both sides using the chain and product rules, collect terms, and solve:

  • Final answer:

Finding Tangent Lines Using Implicit Differentiation

• Given at :

  • Differentiate both sides, collect terms, and solve:

  • At the given point,

  • Equation:

• Given at (1,1):

  • Differentiate:

  • Group :

  • At (1,1):

  • Equation:

Summary Table: Derivatives of Inverse Trigonometric Functions

Additional info: The above notes include expanded explanations, step-by-step differentiation, and summary tables for derivatives of inverse trigonometric functions, as well as detailed examples of implicit differentiation and tangent line calculations. These topics are central to calculus and are typically covered in chapters on derivatives and their applications.