C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M MOVEMENT: Remain in seat during instruction. P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed. NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK. S Activity: Teacher-Directed Instruction

Calculus AB 2013 Implicit Differentiation

Objective C: The swbat differentiate implicitly equations in more than one variable. L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively

Implicit Differentiation Equation for a line: Explicit Form Implicit Form Differentiate the Explicit Differentiation taking place with respect to x. The derivative is explicit also.

Implicit Differentiation Equation of circle: To work explicitly; must work two equations Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly. ( Usually non-functions) Don’t want to solve for y

Implicit Differentiation Chain Rule Pretend y is some function like so becomes (A) (B) (C) Note: Use the Leibniz form. Leads to Parametric and Related Rates. Find the derivative with respect to x

Implicit Differentiation Find the derivative with respect to x

Implicit Differentiation (D) Product Rule

Implicit Differentiation (E) Chain Rule Product inside a chain

Implicit Differentiation (E) Chain Rule Product inside a chain

Implicit Differentiation To find implicitly. EX: Diff Both Sides of equation with respect to x Solve for

C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M MOVEMENT: Remain in seat during instruction. P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed. NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK. S Activity: Teacher-Directed Instruction

Objective C: The swbat differentiate implicitly equations in more than one variable. L: the sw explain to others how to find derivatives of multiple types of problems verbally and demonstratively

EX 1: (a) Find the derivative at the point ( 5, 3 ), at ( -1,-3 ) (b) Find where the curve has a horizontal tangent. (c) Find where the curve has vertical tangents.

EX 1: (b) Find where the curve has a horizontal tangent. Horizontal tangent has a 0 slope

EX 1: (c) Find where the curve has vertical tangents. Vertical tangent has an undefined slope

Ex 2:

Why Implicit? Explicit Form:

Ex 2 Graph: Plot the Folium of Descartes on your graphing calculator and determine the portion of the folium generated when (a) t 0 Parametric Form:

2 nd Derivatives NOTICE:The second derivative is in terms of x, y, AND dy /dx. The final step will be to substitute back the value of dy / dx into the second derivative. EX: Our friendly circle. Find the 2 nd Derivative.

2 nd Derivatives EX: Find the 2 nd Derivative.

Higher Derivatives EX: Find the Third Derivative.

Last update 10/19/10  p – 29 odd