Math 135 Business Calculus Spring 2009Class Notes1.5 Differentiation Techniques: The Power and Sum-Difference Rules

LEIBNIZ NOTATION

So far we’ve been using the notation f 0(x) for the derivative of a function f(x), a notation whichwas introduced by Sir Isaac Newton. The German mathematician Wilhelm Gottfried Leibniz used the

alternative notationdy

dxfor the derivative. Both notations are used today and each has advantages in

certain situations. The derivative of a function y = f(x) can be written in either form

f 0(x) =dy

dx.

The symbold

dxis called a differentiation operator. When placed in front of a function, it means

“the derivative of what comes after it with respect to x.” Therefored

dx

£f(x)

§means “the derivative of f(x) with respect to x.”

For instance,d

dx(x2) means the derivative of x2 with respect to x, so

d

dx(x2) = 2x.

To specify the derivative at a specific number x = a, we write

f 0(a) in Newton’s notation ordy

dx

ØØØØx=a

in Leibniz’ notation

THE POWER RULE

In the previous section, we showed thatd

dx(x2) = 2x and

d

dx(x3) = 3x2.

These are special cases of the following general differentiation rule:

THEOREM 1 The Power RuleFor any real number k,

d

dx(xk) = k · xk−1.

According to the Power Rule, to obtain the derivative of xk we “pull” the exponent out in front andwrite it as a coefficient and subtract 1 from the exponent.

EXAMPLE Differentiate each of the following:a) y = x5

b) y =1x4

c) y =√

x

19

20 Chapter 1 Differentiation

THE DERIVATIVE OF A CONSTANT FUNCTION

Suppose f(x) is a constant function f(x) = ck. The graph of f(x) is a horizontal line. Since a horizontalline has slope 0 at all points, then f 0(x) = 0.

THEOREM 2 Derivative of a ConstantThe derivative of a constant function is 0. That is, for any constant c,

d

dx(c) = 0.

THE DERIVATIVE OF A CONSTANT TIMES A FUNCTION

The limit of a constant multiple of a function equals the contant multiple of the limit. It follows fromthis Limit Principle that the derivative of a constant multiple of a function is the constant multiple ofthe derivative.

THEOREM 3 The Constant Multiple RuleThe derivative of a constant times a function is the constant times the derivative of the function. Forany constant c and differentiable function f(x),

d

dx

£c · f(x)

§= c · d

dx

£f(x)

§.

EXAMPLE Find each of the following derivatives:

a)d

dx(7x4)

b)d

dx

µ2

3x5

∂

EXAMPLE For a spherical tumor, its volume V can be approximated by V (r) = 43πr3, where r is

the radius of the tumor, in centimeters.a) Find the rate of change of the volume with respect to the radius.

b) Find the rate of change of the volume at r = 1.2 cm.

1.5 Differentiation Techniques: The Power and Sum-Difference Rules 21

THE DERIVATIVE OF A SUM OR A DIFFERENCEThe limit of a sum is the sum of the limits. It follows from this Limit Principle that the derivative ofa sum of two functions is the sum of the derivatives, with a similar result for differences.

THEOREM 4 The Sum-Difference RuleFor any differentiable functions f(x) and g(x),Sum. The derivative of a sum is the sum of the derivatives:

d

dx

£f(x) + g(x)

§=

d

dx

£f(x)

§+

d

dx

£f(x)

§.

Difference. The derivative of a difference is the difference of the derivatives:d

dx

£f(x)− g(x)

§=

d

dx

£f(x)

§− d

dx

£f(x)

§.

EXAMPLE Find each of the following derivatives:

a)d

dx5x3 − 7x)

b)d

dx

°24x−

√x +

5x

¢

SLOPES OF TANGENT LINESThe slope of the tangent line to a graph at a point equals the derivative at that point. We can use thisto determine points at which the tangent line has a certain slope and to obtain detailed informationabout the graph.EXAMPLE Find the equation of the tangent line to the graph of f(x) = −1

3x3 + 2x2 at the point(3, 9).

–3 –2 –1 1 2 3 4 5 6 7 8x

–2

–1

1

2

3

4

5

6

7

8

9

10

11

12y

22 Chapter 1 Differentiation

EXAMPLE Find the points on the graph of f(x) = −13x3+2x2 at which the tangent line is horizontal.

–3 –2 –1 1 2 3 4 5 6 7 8x

–2

–1

1

2

3

4

5

6

7

8

9

10

11

12y

EXAMPLE Find the points on the graph of f(x) = −13x3 +2x2 where the tangent line has slope −5.

–3 –2 –1 1 2 3 4 5 6 7 8x

–2

–1

1

2

3

4

5

6

7

8

9

10

11

12y