3.3 Rules for Differentiation

Colorado National Monument

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

0dc

dx

example: 3y

0y

The derivative of a constant is zero.

Consider the function y = 4 graphedto the right. Clearly, the slope of all the tangent lines onthis graph would all be 0.

If we find derivatives with the difference quotient:

2 22

0limh

x h xdx

dx h

2 2 2

0

2limh

x xh h x

h

2x

3 33

0limh

x h xdx

dx h

3 2 2 3 3

0

3 3limh

x x h xh h x

h

23x

2

4dx

dx

4 3 2 2 3 4 4

0

4 6 4limh

x x h x h xh h x

h

34x

2 3

We observe a pattern: 2x 23x 34x 45x 56x …

1n ndx nx

dx

examples:

4f x x

34f x x

8y x

78y x

power rule

We observe a pattern: 2x 23x 34x 45x 56x …

3)( xxf Differentiate

d ducu c

dx dx

examples:

1n ndcx cnx

dx

constant multiple rule:

5 4 47 7 5 35dx x x

dx

When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

(Each term is treated separately)

d ducu c

dx dx

constant multiple rule:

sum and difference rules:

d du dvu v

dx dx dx d du dv

u vdx dx dx

4 12y x x 34 12y x

4 22 2y x x

34 4dy

x xdx

Example:Find the x values of the horizontal tangents of:

4 22 2y x x

34 4dy

x xdx

Horizontal tangents occur when slope = zero.34 4 0x x

3 0x x

2 1 0x x

1 1 0x x x

0, 1, 1x

4 22 2y x x

4 22 2y x x

2y

4 22 2y x x

2y

1y

4 22 2y x x

4 22 2y x x

First derivative (slope) is zero at:

0, 1, 1x

34 4dy

x xdx

product rule:

Notice that this is not just the product of two derivatives.

dx

dvuv

dx

duuv

dx

d)(

)52)(3()( 32 xxxxh

Since multiplication is commutative, the product rule order can change. The book uses this one.

I do not.

another way to remembered it :

467)( 3 xxxh

f(x) g(x)

so )()()()()( xgxfxgxfxh

der. of 1st times the 2nd + 1st times the der. of the 2nd

68421

167421)(23

32

xx

xxxxh

d dv duuv u v

dx dx dx

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

Since it is subtraction, the order does matter

2)(

)()()()()(

xg

xgxfxgxfxh

4

62

)(

)()(

2

4

x

xx

xg

xfxh

Same as product, but subtraction in thenumerator

22

423

4

)2)(62()4)(68()(

x

xxxxxxh

Find the equation of the tangent line to the curve

What is the equation of the normal line to f(x) = at x = –4? x

5

What is the velocity as a function of time (in seconds) of the time-position function meters?

5

3 4)(

t

ttts

Example: working with numerical values

Let y = uv be the product of the functions u and v. Find y’(2) if

2)2(

,1)2(

,4)2(

,3)2(

v

v

u

u

Ans: Find the product

y’ = u’ v + u v’ y’(2)= (-4)(1)+(3)(2) = -4+6 = 2

Higher Order Derivatives:

dyy

dx is the first derivative of y with respect to x.

2

2

dy d dy d yy

dx dx dx dx

is the second derivative.

(y double prime)

dyy

dx

is the third derivative.

4 dy y

dx is the fourth derivative.

We will learn later what these higher order derivatives are used for.

Ex. 1864x yfor Find 23)4( xxy

the end