1. Calculus I (MAT 145) Dr. Day Monday Oct 9, 2017
Derivative Shortcuts (Chapter 3)
Derivatives of Composite Functions (3.4)
Derivatives of Implicitly Defined Functions (3.5)
Monday, October 9, 2017
MAT 145

2. Here's a function in a function And your job here is to find
THE CHAIN RULE
WORDS BY: JOHN A. CARTER
TUNE: "CLEMENTINE"
Here's a function in a function
And your job here is to find
The derivative of the whole thing
With respect to x inside.
Call the outside f of u
And call the inside u of x.
Differentiate to find df/du
And multiply by du/dx.
Use the chain rule.
Use the chain rule when e'er you find
The derivative of a function compositionally defined.
Monday, October 9, 2017
MAT 145

3. Composite Functions
Monday, October 9, 2017
MAT 145

4. Derivatives of Composite Functions (3.4)
Monday, October 9, 2017
MAT 145

5. Derivatives of Composite Functions
Monday, October 9, 2017
MAT 145

6. Using Derivative Patterns
For s(t) = cos(2t):
Calculate s’(t) and s’’(t).
Determine an equation for the line tangent to the graph of s when t = π/8.
Determine the two values of t closest to t = 0 that lead to horizontal tangent lines.
Determine the smallest positive value of t for which s’(t) = 1.
If s(t) represents an object’s position on the number line at time t (s in feet, t in minutes), calculate the object’s velocity and acceleration at time t = π/12. Based on those results, describe everything you can about the object’s movement at that instant.
Monday, October 9, 2017
MAT 145

7. Derivatives of Implicitly Defined Functions (3.5)
An implicitly defined function is:
A function whose relation among the variable is given by an equation for which the function has not been explicitly stated.
In the equation x2 + y2 = 25, y is an implicit function of x because the equation doesn’t explicitly express y in terms of x.
Monday, October 9, 2017
MAT 145

8. Monday, October 9, 2017
MAT 145

9. Implicit Differentiation
Find dy/dx when
Take derivative of each term. Use chain rule.
X changes identically to itself; dx/dx=1.
Solve for dy/dx.
Simplify.
Monday, October 9, 2017
MAT 145

10. Derivatives of Implicitly Defined Functions (3.5)
To calculate the derivative of an implicitly defined function:
Assume a functional connection among the variables. If x and y are present, assume y is a function of x.
Calculate the derivative of each term in the equation. Because we don’t explicitly know how y is determined by x, when calculating the derivative of y, we use the chain rule and write dy/dx as the derivative of y.
After determining term-by-term derivatives, carry out all necessary algebra steps to isolate dy/dx. That’s our goal!
Monday, October 9, 2017
MAT 145

11. Implicit Differentiation
Find dy/dx.
Monday, October 9, 2017
MAT 145

12. Practice Derivative Rules
Monday, October 9, 2017
MAT 145