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Calculus I (MAT 145) Dr. Day Mon day March 4, 2013

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- Implicit Differentiation (3.5)
- Derivatives Involving Logarithms (3.6)
- Logarithmic Differentiation (3.6)
- Assignments

MAT 145

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.

MAT 145

MAT 145

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.

Use the chain rule whene'er you find

The derivative of a function compositionally defined.

MAT 145

MAT 145

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.

MAT 145

- 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 isolatedy/dx. That’s our goal!

MAT 145

MAT 145

Try These:

MAT 145

What is the derivative of the natural log function y = ln(x)?

MAT 145

Now apply this to other log functions:

MAT 145

And extend this to other functions:

MAT 145

WebAssign

- 3.4 (Part 2), 3.5, 3.6 (two parts) assignments up for completion.
- Gateway Derivatives Quiz #2 returned today. GDQ #3 next week!
- Test #3: Friday, March 8.

MAT 145