Clicker Question 1

What is the derivative of f (x ) = x 3 ex ? A. 3x 2 ex

B. ex (x 3 + 3x 2) C. ex (x 3 – 3x 2) D. 3x 3ex – 1

E. x 4ex – 1 + 3x 2ex

Clicker Question 2

What is the instantaneous rate of change of g (x ) = x /(x – 5) at x = 2? A. 1 B. -5/9 C. -5/(x – 5)2

D. 5/9 E. 5/(x – 5)2

Derivatives of Trig Functions (11/29/10) It is easy to see by sketching that the

derivative of the sine function looks an awful lot like the cosine function and the derivative of the cosine function an awful lot like the negative sine function.

This is in fact the case. The proof of the first is similar to the

case of exponential functions, but requires use of the sin(A + B) formula.

Some old trig formulas

Sine of a sum:sin(A+B) = sin(A)cos(B) + sin(B)cos(A)

So if f (t ) = sin(t), then we get f '(t ) =

Now group and simplify.h

ththth

)sin()sin()cos()cos()sin(lim 0

Working out the Sine Upon working through the definition of the

derivative of the f (t) = sin(t), we get f '(t) = sin(t) * (the derivative of cos at 0) + cos(t) * (the derivative of sin at 0)

Those two numbers are, respectively, 0 and 1, and so we are done.

Note the similarity to the derivative of a x.

Clicker Question 3

What is the slope of the tangent line to the curve y = sin(x ) at the point (/6, ½)? A. 3 / 2 B. 1 / 2 C. cos(x ) D. 1 E. /6

An aside: Another notation for f '(x)

The notation d/dx is often used to denote the derivative of a function of x.

If the function is denoted by y , we write dy/dx , as opposed to y '.

The top tells you the name of the output, the bottom the input variable, and the d’s denote “change in.”

Example: d/dx (x 4) = 4x 3

Derivative of the Tangent

Since the tangent function is defined as a quotient, guess what rule we can use to compute its derivative?

We see that d(tan(t ))/dt = 1/cos2(t ) = sec2(t )

Note that it makes sense that this derivative function is always positive. (Why?)

Assignment for Wednesday

Read Section 3.3 and do Exercises 1, 5, 9 13, 18, 23, 25 and 33.

Hand-in #4 is due Thursday (12/2) at 4:45.