consider the function
DESCRIPTION
slope. Consider the function. We could make a graph of the slope:. Now we connect the dots!. The resulting curve is a cosine curve. slope. We can do the same thing for. The resulting curve is a sine curve that has been reflected about the x-axis. Derivative of y=sinx. - PowerPoint PPT PresentationTRANSCRIPT
20
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
20
2
We can do the same thing for cosy slope
0
1
0
1
0The resulting curve is a sine curve that has been reflected about the x-axis.
cos sind x xdx
Derivative of y=sinx
• Use the definition of the derivative
To prove the derivative of y=sinx is y’=cosx.
lim ( ) ( )'0f x h f xy
h h
Hints:sin( ) sin cosh cos sinh
lim cosh 1 00
x h x x
h h
Derivative of y=sinxlim ( ) ( )'
0f x h f xy
h h
lim sin( ) sin0
x h xh h
sin 0 cos 1x x lim sin cosh cos sinh sin0
x x xh h
lim sin (cosh 1) cos sinh0
x xh h
lim lim lim limcosh 1 sinhsin cos0 0 0 0
x xh h h hh h
lim cosh 1 sinhsin cos0
x xh h h
Shortcut: y’=cosx
The proof of the d(cosx) = -sinx is almost identical
product rule:
d dv duuv u vdx dx dx
Notice that this is not just the product of two derivatives.
This is sometimes memorized as: d uv u dv v du
2 33 2 5d x x xdx
5 3 32 5 6 15d x x x xdx
5 32 11 15d x x xdx
4 210 33 15x x
2 3x 26 5x 32 5x x 2x
4 2 2 4 26 5 18 15 4 10x x x x x
4 210 33 15x x
Example:
2 cosd x xdx
2
2 coscosdx d xx xdx dx
2 cos2 cos d xx x xdx
22 cos sinx x x x
Try this:
21. sind xdx
2sin sin sind dx x xdx dx
sin sin sin sind dx x x xdx dx
2sin cosx x
Try this:
2 22. sin cosd x xdx
1 0ddx
2 2
Or,
sin cosd dx xdx dx
2sin cos
cos cos cos cos
x xd dx x x xdx dx
2sin cos
sin cos cos sinx x
x x x x
2sin cos 2sin cos x x x x 0
quotient rule:
2
du dvv ud u dx dxdx v v
or 2
u v du u dvdv v
3
2
2 53
d x xdx x
2 2 3
22
3 6 5 2 5 2
3
x x x x x
x
Quotient Rule
2
1 sin5
d xdx x
2 2
22
1 sin 5 1 sin 5
5
x x x x
x
2
22
5 cos 1 sin 10
5
x x x x
x
3
cos 2sin 25
x x xx
1cos
ddx x
2
1 cos 1 cos
cos
d dx xdx dx
x
2
sincos
xx
1 sincos cos
xx x
sec tanx x