The composition The composition functionfunction

k(x) = (k(x) = (ff o o g)(x) = g)(x) = ff (g(x)) (g(x))

g g ff

R->[0,+oo)->[-1, 1] R->[0,+oo)->[-1, 1]

Theorem 1 Theorem 1 The chain ruleThe chain rule

g g ff

R-->[0,+oo)-->[-1, 1]R-->[0,+oo)-->[-1, 1]

If k(x) =If k(x) =ff (g(x)), then (g(x)), then

k’(x)k’(x) = = f ’f ’ ( ( g(x)g(x) ) ) g’(x)g’(x)

If k(x) =If k(x) =ff (g(x)), then (g(x)), thenk’(x)k’(x) = = f ’f ’ (( g(x)g(x) )) g’(x)g’(x)k(x) = k(x) = sinsin( x( x22 ) )

k’(x)k’(x) = = cos cos ( x( x2 2 ) ) 2x2x

If y=sin(3If y=sin(3t), find y’t), find y’

A.A. 33 cos(3 cos(3t)t)

B.B. cos(t)cos(t)

C.C. 33t[cos(3t[cos(3t)]t)]

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If y=sin(3If y=sin(3t), find y’t), find y’

A.A. 33 cos(3 cos(3t)t)

B.B. cos(3cos(3t)t)

C.C. 33t[cos(3t[cos(3t)]t)]

If y=sin(3If y=sin(3t), find y’t), find y’

A.A. 33 cos(3 cos(3t)t)

B.B. cos(3cos(3))

C.C. 33t[cos(3t[cos(3t)]t)]

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If y=sin(3pt), find y’If y=sin(3pt), find y’

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A.

B.

C.

3p cos(3pt)

cos(3p)

3pt[cos(3pt)]

33.3%

66.7%

100.0%

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If y=sin(3If y=sin(3t), find y’t), find y’

A.A. 33 cos(3 cos(3t)t)

B.B. cos(3cos(3t)t)

C.C. 33t[cos(3t[cos(3t)]t)]

CorallaryCorallary

k(x) = gk(x) = gnn(x) = [g(x)](x) = [g(x)]nn

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

y=(2x+1)y=(2x+1)44, find y’, find y’

A.A. 4(2)^34(2)^3

B.B. 4(2x+1)^34(2x+1)^3

C.C. 8(2x+1)8(2x+1)

D.D. 8(2x+1)^38(2x+1)^3

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y=(2x+1)4, find y’y=(2x+1)4, find y’

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B.

C.

D.

4(2)^3

4(2x+1)^3

8(2x+1)

8(2x+1)^3

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If y=(2x+1)If y=(2x+1)44, find y’, find y’

A.A. 4(2)4(2)33

B.B. 4(2x+1)4(2x+1)33

C.C. 8(2x+1)8(2x+1)

D.D. 8(2x+1)8(2x+1)33

If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’

A.A. -sec-sec22[sin(x)]cos(x)[sin(x)]cos(x)

B.B. secsec22[sin(x)]cos(x)[sin(x)]cos(x)

C.C. secsec22[cos(x)][cos(x)]

D.D. -csc-csc22[sin(x)]cos(x)[sin(x)]cos(x)

If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’

A.A. 11

B.B. 22

C.C. 33

D.D. 44

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If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’

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If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’

A.A. -sec-sec22[sin(x)]cos(x)[sin(x)]cos(x)

B.B. secsec22[sin(x)]cos(x)[sin(x)]cos(x)

C.C. secsec22[cos(x)][cos(x)]

D.D. -csc-csc22[sin(x)]cos(x)[sin(x)]cos(x)

If y=x cos(xIf y=x cos(x22), find ), find dy/dxdy/dx

A.A. -x sin(x-x sin(x22) + cos(x) + cos(x22))

B.B. -2x sin(x-2x sin(x22) + cos(x) + cos(x22))

C.C. -2x-2x22 sin(x sin(x22) + cos(x) + cos(x22))

D.D. 2x2x22 sin(x sin(x22) + cos(x) + cos(x22))

If y=x cos(xIf y=x cos(x22), find ), find dy/dxdy/dx

A.A. 11

B.B. 22

C.C. 33

D.D. 44

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If y=x cos(x2), find If y=x cos(x2), find dy/dxdy/dx

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If y=x cos(xIf y=x cos(x22), find ), find dy/dxdy/dx

A.A. -x sin(x-x sin(x22) + cos(x) + cos(x22))

B.B. -2x sin(x-2x sin(x22) + cos(x) + cos(x22))

C.C. -2x-2x22 sin(x sin(x22) + cos(x) + cos(x22))

D.D. 2x2x22 sin(x sin(x22) + cos(x) + cos(x22))

The chain ruleThe chain ruleIf y = If y = sin(sin(uu)) and u(x) = x and u(x) = x22

then dy/dx = dy/du du/dxthen dy/dx = dy/du du/dx

dy/du = cos(dy/du = cos(uu) d) duu/dx = /dx = 2x2x

dy/dx = dy/dx = cos(cos(uu) ) 2x2x

= = cos(cos(xx22) ) 2x2x

The chain ruleThe chain ruleIf y = If y = cos(cos(uu)) and u(x) = x and u(x) = x22 + 3x + 3x

then dy/dx = dy/du du/dxthen dy/dx = dy/du du/dx

dy/du = -sin(dy/du = -sin(uu) d) duu/dx = /dx = 2x + 32x + 3

dy/dx = dy/dx = -sin(-sin(uu) ) (2x+3)(2x+3)

= = -sin(-sin(xx22+2x+2x) ) (2x+3)(2x+3)

y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx

A.A. -10 csc-10 csc22(10x-5)(10x-5)

B.B. secsec22(10)(10)

C.C. -csc-csc22(10x-5)(10x-5)

D.D. 10 sec10 sec22(10x-5)(10x-5)

y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx

A.A. 11

B.B. 22

C.C. 33

D.D. 44

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y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx

A.

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y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx

A.A. -10 csc-10 csc22(10x-5)(10x-5)

B.B. secsec22(10)(10)

C.C. -csc-csc22(10x-5)(10x-5)

D.D. 10 sec10 sec22(10x-5)(10x-5)

CorallaryCorallary

k(x) = [3xk(x) = [3x3 3 - x- x-2 -2 ]]2020

k’(x)k’(x) = 20 [3x = 20 [3x3 3 - x- x-2-2] ] 1919 ( (9x9x22+2x+2x--

33))

If y = (sec(x))If y = (sec(x))22=sec=sec22(x) (x) find dy/dxfind dy/dx

A.A. 2 sec(x) tan(x)2 sec(x) tan(x)

B.B. 2 sec2 sec22(x) tan(x)(x) tan(x)

C.C. 2 sec(x)2 sec(x) tantan22(x)(x)

D.D. secsec22(x)(x) tantan (x)(x)

If y = (sec(x))If y = (sec(x))22=sec=sec2 2 (x) (x) find dy/dxfind dy/dx

A.A. 11

B.B. 22

C.C. 33

D.D. 44

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If y = (sec(x))2=sec2 If y = (sec(x))2=sec2 (x) find dy/dx(x) find dy/dx

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If y = (sec(x))If y = (sec(x))22=sec=sec22(x) (x) find dy/dxfind dy/dx

A.A. 2 sec(x) tan(x)2 sec(x) tan(x)

B.B. 2 sec2 sec22(x) tan(x)(x) tan(x)

C.C. 2 sec(x)2 sec(x) tantan22(x)(x)

D.D. secsec22(x)(x) tantan (x)(x)

CorallaryCorallary

=[3x=[3x3 3 - x- x2 2 ]]1/21/2

k’(x)k’(x) = ½ [3x = ½ [3x3 3 - x- x22]]-1/2-1/2 ( (9x9x22-2x-2x))

3 2( ) 3k x x x

CorallaryCorallary

k’(x)k’(x) = =

( ) ( )k x g x

1'( )

2 ( )g x

g x'( )

2 ( )

g x

g x

CorallaryCorallary

k’(x)k’(x) = =

3 2( ) 3k x x x 29x -2x3 22 3x x

If y = find If y = find dy/dx dy/dx

A.A. ..

B.B. ..

C.C. ..

csc( )x

csc( )cot( )

csc( )

x x

x

csc( )cot( )

2 csc( )

x x

x

csc( )cot( )

2 csc( )

x x

x

If y = find If y = find dy/dx dy/dx

A.A. ..

B.B. ..

C.C. ..

csc( )x

csc( )cot( )

csc( )

x x

x

csc( )cot( )

2 csc( )

x x

x

csc( )cot( )

2 csc( )

x x

x

CorallaryCorallary

= [sin(2x)= [sin(2x) ]]1/21/2

k’(x)k’(x) = ½ [sin(2x)] = ½ [sin(2x)]-1/2-1/2 ( (cos(2x) cos(2x) 22))

( ) sin(2 )k x x

k(x) = sec(sin(2x)) k(x) = sec(sin(2x))

k’(x)k’(x) = =

secsec(sin(2x))(sin(2x))tantan(sin(2x))(sin(2x))((cos(2x) 2cos(2x) 2))

quizzquizz

1.Write the equation of the 1.Write the equation of the line tangent to the graph of line tangent to the graph of

y = x – cos(x) when x=0.y = x – cos(x) when x=0.

2. Diff. g(x)=cot x[sin x – cos 2. Diff. g(x)=cot x[sin x – cos x].x].

3. Find the x’s where the lines 3. Find the x’s where the lines tangent to y= are tangent to y= are horizontal.horizontal.

sin

1 cos

x

x