Upload
angel-archer
View
220
Download
0
Tags:
Embed Size (px)
Citation preview
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!The resulting curve is a cosine curve.
sin cosd
x xdx
2.3 Derivatives of Trigonometric Functions
2.3 Derivatives of Trigonometric Functions
h
xhxx
dx
dh
sin)sin(limsin
0
h
xxhhxx
dx
dh
sincossincossinlimsin
0
h
xh
h
hxx
dx
dhh
cossinlim
)1(cossinlimsin
00
h
xhhxx
dx
dh
cossin)1(cossinlimsin
0
Proof
2.3 Derivatives of Trigonometric Functions
h
xh
h
hxx
dx
dhh
cossinlim
)1(cossinlimsin
00
= 0 = 1
sin cosd
x xdx
2.3 Derivatives of Trigonometric Functions
h
xhxx
dx
dh
cos)cos(limcos
0
h
xxhhxx
dx
dh
cossinsincoscoslimcos
0
h
xh
h
hxx
dx
dhh
sinsinlim
)1(coscoslimcos
00
h
xhhxx
dx
dh
sinsin)1(coscoslimcos
0
Find the derivative of cos x
2.3 Derivatives of Trigonometric Functions
= 0 = 1
h
xh
h
hxx
dx
dhh
sinsinlim
)1(coscoslimcos
00
cos sind
x xdx
We can find the derivative of tangent x by using the quotient rule.
tand
xdx
sin
cos
d x
dx x
2
cos cos sin sin
cos
x x x x
x
2 2
2
cos sin
cos
x x
x
2
1
cos x
2sec x
2tan secd
x xdx
2.3 Derivatives of Trigonometric Functions
=
=
=
=
=
Derivatives of the remaining trig functions can be determined the same way.
sin cosd
x xdx
cos sind
x xdx
2tan secd
x xdx
2cot cscd
x xdx
sec sec tand
x x xdx
csc csc cotd
x x xdx
2.3 Derivatives of Trigonometric Functions
2.3 Derivatives of Trigonometric Functions
dy dy du
dx du dx Chain Rule:
2.4 Chain Rule
Let h(x) = f(g(x)) (also known as )
Then h’(x) = (f(g(x))’ =
f g
at at xu g xf g f g
)('))((' xgxgf
Here is a faster way to find the derivative:
2sin 4y x
2 2cos 4 4d
y x xdx
2cos 4 2y x x
Differentiate the outside function...
…then the inside function
2.4 Chain Rule
‘
‘
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
2cos 3 sin 3 3d
x x xdx
2cos 3 sin 3 3x x
6cos 3 sin 3x x
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)
2.4 Chain Rule
=
=
=
=
=
Derivative formulas include the chain rule!
1n nd duu nu
dx dx sin cos
d duu u
dx dx
cos sind du
u udx dx
2tan secd du
u udx dx
etcetera…
2.4 Chain Rule
2.4 Chain Rule
Find
)3cos( 2 xxy )16)(3sin( 2 xxxdx
dy
))sin(cos(xy
)24(cos 33 xxy
)sin)(cos(cos xxdx
dy
)212))(24sin()(24(cos3 2332 xxxxxdx
dy
))24sin()(24(cos)636( 3322 xxxxxdx
dy
dx
dy