Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

AP Warmup #2

1) Suppose the derivative of the function f(x) exists at x = a. Which one of the following expressions is NOT equal to the derivative of f at a?

f (a  h)  f (a) f (a)  f (a  h) lim lim (a) h (b) h h0 h0

f (a  h)  f (a) f (a)  f (a  h) lim lim (c) h (d)  h h0 h0

f (a)  f (b) lim (e) a  b ba

The Product Rule…..In Your Terms

Some Examples: Find the derivative.

(1) f( x ) (5 x23  1)(2 x  2) (2) g( t ) tan t  (sin t  5 t )

Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

The Quotient Rule…..In Your Terms

31x2  (3) fx() 35x 

x 1 1 (4) Find an equation of a line tangent to fx() at the point 2, . x 1 3

One more Trig Example

1 cos x Given fx() , find fx()……Ohhhh, the places you’ll go with this one! sin x

Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

What is the Chain Rule and Why Do We Need It?

It gives us a way to find the derivative of a composite function! A who? You know, something like….. fg( (x)) . What would that even look like? Glad you asked!

Suppose f() x x and g( x ) x2 1 . Then, f( g (x)) x2 1

Outside Inside function function s s

But how do we derive this?

1  If not done already, rewrite as the “inside function” to a power (x2  1) 2

 Derive the outside function using the power rule  1 1 (x2  1) 2 o Do you notice – we DO NOT change the inside function 2

 1  Multiply your result by the derivative of the inside function 1 (x2  1) 2 [2x ] 2

 Simplify if possible x 2 x 1 Definition of the Chain Rule

If dealing with graphs or tables: If dealing with actual functions: dy dy du  Take the derivative of the outside, dx du dx times the derivative of the inside OR d f[ g ( x )] f ( g ( x )) g ( x ) dx

Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

Some Examples: Find the derivative.

(1) f( x ) (3 x 2 x23 ) (2) g( x ) cos2 x

22 (3) q( w ) w 1 w (4) f (x) cos(3x)

(5) f( x ) csc(4 x2 3 x ) (6) fx(y) tan cos(5 )

Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

Let’s take a look Graphically!

Setting the TABLE!

Let h( x ) f ( g ( x )) . Fill in the table below so that such that the values of h(0) and h(5) can be found.

h(0)  h(5) 

Unit 2 - Differentiation Lesson 2: Differentiation Techniques Product, Quotient, & Chain Rules

Extending the Idea

Assume f( x ) ( x22  1)( x  1) . How many horizontal tangents of f exist?

Use the table below to answer #1-6.

Determine the value of:

d x  3 d f() x x  2 1) [f ( x ) g ( x )] at 2)  at dx dx g() x

d d 3) [f ( g ( x ))] at 4) [()]fx at dx dx

d 1 22 5)  at x  3 6) If h()()() x f x g x dx g() x

then find h(2) .