Calculus and Analytical Geometry MTH 104 Lecture # 10 Calculus and Analytical Geometry

Explicit and Implicit Functions An equation of the form is said to define . For example explicitly as a function of Implicit Function

Example Use implicit differentiation to find if Solution

Implicit differentiation To find Differentiate both sides with respect to x, creating y as a differentiable function of x Solve for Example Find of

Example Use implicit differentiation to find of Solution Again differentiating both side implicitly

Example Find the slopes of the tangent line to the curve at the points (2, -1) and (2, 1). Solution

Slope of the tangent line at (2, -1)

Derivatives of logarithmic functions Generalized derivative formulas

Examples Find Solution 1.

2. Product rule 3.

Product rule

4. 5. Example Find using logarithmic differentiation

Solution Taking ln of both sides Differentiating with respect to

Derivatives of exponential Generalized form

Examples 1. 2. 3. 4.

Derivatives of inverse Trigonometric Functions Generalized derivative formulas 1. 2. 3. 4. 5.

6. Example Find if solution

Example Find if solution