Clicker Question 1 What is the derivative of f (x ) = e3x sin(4x ) ?

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3.7 Implicit Differentiation

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Presentation transcript:

Clicker Question 1 What is the derivative of f (x ) = e3x sin(4x ) ? A. e3x cos(4x ) B. e3x (cos(4x ) + sin(4x )) C. e3x (4cos(4x ) + 3sin(4x )) D. e3x (3cos(4x ) + 4sin(4x )) E. 12e3x cos(4x )

Clicker Question 2 What is an equation of the tangent line to the curve y = (x 2 – 1)5 at the point (0, -1)? A. y = -1 B. y = 5x – 1 C. y = -5x – 1 D. y = 5(x 2 – 1)4 – 1 E. y = 10x (x 2 – 1)4 – 1

Using the Chain Rule: Implicit Differentiation (3/16/09) Sometime functions are defined “implicitly” rather than explicitly, i.e., the independent and dependent variables are mixed together in an equation, as opposed to simply being solved for the dependent variable. We can use the Chain Rule to find the derivative implicitly, viewing y as a “chunk”.

Implicit Differentiation: The procedure Given an equation with x (the input, or independent variable) and y (the output, or dependent variable) mixed together: Take the derivative of every term with respect to x , remembering that y is the “chunk”, so you must use the Chain Rule. Solve the resulting equation for y '

Implicit Differentiation: An example Given x 2 + y 2 = 4 (what is this?), find dy / dx two ways: Explicitly, i.e., first solve for y . Implicitly, i.e., take the derivative of both sides with respect to x , treating y as a “chunk”. Check that this answer makes sense! For example, what’s the slope of the tangent line to the curve at (2, 2)? At (-2, 2)? At (0, 2)? At (2, 0)?

Clicker Question 3 What is y ' if 4x 3 + 5y 2 = 7 ? A. 7 – 12x 2 – 10 y B. 42x 2 / 5y C. -12x 2 – 10y D. 6x 2 / 5y E. -6x 2 / 5y

Implicit Differentiation: An upside and a downside If y is given implicitly (i.e., the x ‘s and y ‘s are mixed together): The upside of implicit differentiation is you can move right ahead on differentiating to start with, using the Chain Rule. The downside is that when you solve for y ‘, the x ‘s and y ‘s are still mixed together.

Assignment for Wednesday Read Section 3.5. Do Exercises 1-19 odd, 25, 29, and 39.