consider the function

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