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 h0 h0 f (a h) f (a) f (a) f (a h) lim lim (c) h (d) h h0 h0 f (a) f (b) lim (e) a b ba 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. 23 2 (1) f( x ) (3 x 2 x ) (2) g( x ) cos 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) . .