If k(x) =f (g(x)), then k’(x) = f ’ ( g(x) ) g’(x) k(x) = sin( x 2 ) k’(x) = cos ( x 2 ) 2x.

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Presentation transcript:

If k(x) =f (g(x)), then k’(x) = f ’ ( g(x) ) g’(x) k(x) = sin( x 2 ) k’(x) = cos ( x 2 ) 2x

If y = sec(3  t), find y’ A. 3  sec(3  t) tan(3  t) B. 3  sec tan (3  t) C. sec(3  t) tan(3  t)

If y = sec(3  t), find y’ A. 3  sec(3  t) tan(3  t) B. 3  sec tan (3  t) C. sec(3  t) tan(3  t)

If y=tan(sin(x)), find y’ A. -sec 2 [sin(x)]cos(x) B. sec 2 [sin(x)]cos(x) C. sec 2 [cos(x)] D. -csc 2 [sin(x)]cos(x)

If y=tan(sin(x)), find y’ A. -sec 2 [sin(x)]cos(x) B. sec 2 [sin(x)]cos(x) C. sec 2 [cos(x)] D. -csc 2 [sin(x)]cos(x)

Corallary k(x) = g n (x) = [g(x)] n k’(x) = n [g (x)] n-1 g’(x)

If y=(2x+1) 4, find y’ A. 4(2) 3 B. 4(2x+1) 3 C. 8(2x+1) D. 8(2x+1) 3

If y=(2x+1) 4, find y’ A. 4(2) 3 B. 4(2x+1) 3 C. 8(2x+1) D. 8(2x+1) 3

If y=x cos(x 2 ), find dy/dx A. -x sin(x 2 ) + cos(x 2 ) B. -2x sin(x 2 ) + cos(x 2 ) C. -2x 2 sin(x 2 ) + cos(x 2 ) D. 2x 2 sin(x 2 ) + cos(x 2 )

If y=x cos(x 2 ), find dy/dx A. -x sin(x 2 ) + cos(x 2 ) B. -2x sin(x 2 ) + cos(x 2 ) C. -2x 2 sin(x 2 ) + cos(x 2 ) D. 2x 2 sin(x 2 ) + cos(x 2 )

The chain rule If y = sin(u) and u(x) = x 2 then dy/dx = dy/du du/dx dy/du = cos(u) du/dx = 2x dy/dx = cos(u) 2x = cos(x 2 ) 2x

The chain rule If y = cos(u) and u(x) = x 2 + 3x then dy/dx = dy/du du/dx dy/du = -sin(u) du/dx = 2x + 3 dy/dx = -sin(u) (2x+3) = -sin(x 2 +2x) (2x+3)

y=tan(u) u = 10x – 5 find dy/dx A. -10 csc 2 (10x-5) B. sec 2 (10) C. -csc 2 (10x-5) D. 10 sec 2 (10x-5)

y=tan(u) u = 10x – 5 find dy/dx A. -10 csc 2 (10x-5) B. sec 2 (10) C. -csc 2 (10x-5) D. 10 sec 2 (10x-5)

y= u 2 +u u = 10x 2 – x find dy/dx A. (20 x 2 – 2x)(20x-1) B. (20 x 2 – 2x +1)20x C. (20 x 2 - 1)(20x-1) D. (20 x 2 – 2x +1)(20x-1)

y= u 2 +u u = 10x 2 – x find dy/dx A. (20 x 2 – 2x)(20x-1) B. (20 x 2 – 2x +1)20x C. (20 x 2 - 1)(20x-1) D. (20 x 2 – 2x +1)(20x-1)

Corallary k(x) = [3x 3 - x -2 ] 20 k’(x) = 20 [3x 3 - x -2 ] 19 (9x 2 +2x -3 )

Corallary y = [3x 3 - x -2 ] 20 let u = [3x 3 - x -2 ] let u = [3x 3 - x -2 ] du/dx = (9x 2 +2x -3 ) y=u 20 dy/dx=dy/du du/dx = 20u 19 du/dx du/dx = (9x 2 +2x -3 ) y=u 20 dy/dx=dy/du du/dx = 20u 19 du/dx = 20 [3x 3 - x -2 ] 19 (9x 2 +2x -3 ) = 20 [3x 3 - x -2 ] 19 (9x 2 +2x -3 )

If y = (sec(x)) 2 =sec 2 (x) find dy/dx A. 2 sec(x) tan(x) B. 2 sec 2 (x) tan(x) C. 2 sec(x) tan 2 (x) D. sec 2 (x) tan (x)

If y = (sec(x)) 2 =sec 2 (x) find dy/dx A. 2 sec(x) tan(x) B. 2 sec 2 (x) tan(x) C. 2 sec(x) tan 2 (x) D. sec 2 (x) tan (x)

Corallary =[3x 3 - x 2 ] 1/2 =[3x 3 - x 2 ] 1/2 k’(x) = ½ [3x 3 - x 2 ] -1/2 (9x 2 -2x)

Corallary k’(x) =

Corallary

If y = find dy/dx A. csc 3/2 (x) B.. C.. D..

If y = find dy/dx A. csc 3/2 (x) B.. C.. D..

Corallary = [sin(2x) ] 1/2 = [sin(2x) ] 1/2 k’(x) = ½ [sin(2x)] -1/2 (cos(2x) 2)

k(x) = sec(sin(2x)) k’(x) = sec(sin(2x))tan(sin(2x))(cos(2x) 2)

y = sec(sin(2x)) let u = sin(2x) dy/dx = dy/du du/dx y = sec u

y = sec(u) where u = sin(2x) dy/dx = dy/du du/dx = sec u tan u cos(2x) 2

y = sec(u) where u = sin(2x) dy/dx = dy/du du/dx = sec u tan u cos(2x) 2 sec(sin(2x))tan(sin(2x))(cos(2x) 2)

Number of heart beats per minute, t seconds after the beginning of a race is given by a) Find and explain. b) Find R’(t). c) Find R’(10) and explain. d) Find R(10) and explain.

Number of heart beats per minute, t seconds a) Find and explain.

Number of heart beats per minute, t seconds a) Find and explain.

Number of heart beats per minute, t seconds a) Find and explain.

Number of heart beats per minute, t seconds a) Find and explain. Mary’s maximum heart rate is 200 bpm = 220 – age making her age close to 20.

Number of heart beats per minute, t seconds after the beginning of a race is given by a). b) Find R’(t) c) Find R(10) = bpm d) Find R’(10) and explain.

Number of heart beats per minute, t seconds Find R’(t)

Number of heart beats per minute, t seconds Find R’(t) R’(10) = bpm/min

quizz 1.Write the equation of the line tangent to the graph of y = x – cos(x) when x=0. 2. Diff. g(x)=cot x [sin x – cos x]. 3. Find the x’s where the lines tangent to y= are horizontal.