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

0

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 ( ) ( )'

0

f x h f xy

h h

Hints:

sin( ) sin cosh cos sinh

lim cosh 10

0

x h x x

h h

Derivative of y=sinx

lim ( ) ( )'

0

f x h f xy

h h

lim sin( ) sin

0

x h x

h h

sin 0 cos 1x x lim sin cosh cos sinh sin

0

x x x

h h

lim sin (cosh 1) cos sinh

0

x x

h h

lim lim lim limcosh 1 sinhsin cos

0 0 0 0x x

h h h hh h

lim cosh 1 sinhsin cos

0x x

h h h

Shortcut: y’=cosx

The proof of the d(cosx) = -sinx is almost identical

http://www.calculus-help.com/the-product-rule/

product rule:

d dv duuv u v

dx 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 cosdx x

dx

22 cos

cosdx d x

x xdx dx

2 cos2 cos

d xx x x

dx

22 cos sinx x x x

Try this:

21. sind

xdx

2sin sin sind d

x x xdx dx

sin sin sin sind d

x x x xdx dx

2sin cosx x

Try this:

2 22. sin cosd

x xdx

1 0d

dx

2 2

Or,

sin cosd d

x xdx dx

2sin cos

cos cos cos cos

x x

d dx x x x

dx dx

2sin cos

sin cos cos sin

x x

x x x x

2sin cos 2sin cos x x x x 0

http://www.calculus-help.com/the-quotient-rule/

quotient rule:

2

du dvv ud u dx dx

dx v v

or 2

u v du u dvdv v

3

2

2 5

3

d x x

dx x

2 2 3

22

3 6 5 2 5 2

3

x x x x x

x

Quotient Rule

2

1 sin

5

d x

dx 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 2

5

x x x

x

1

cos

d

dx x

2

1 cos 1 cos

cos

d dx x

dx dxx

2

sin

cos

x

x

1 sin

cos cos

x

x x

sec tanx x