4.1: Techniques for DifferentiationObjectives: Students will be able to…

• Apply the constant and power rule for differentiation

• Apply the sum and difference rule for differentiation

•The definition and interpretation of the derivative will NOT change, but we do have rules that make finding the derivative easier.

•Pay attention to variables used. If , the derivative would be g ‘(t) or

•Another notation for the derivative: )]([ xfDx

1)( 2 ttg

dt

dg

What is the slope of the function f(x) = 10?

f(x) = -5?? f(x)= 15,000??

Constant slope, constant rate of changeSlope always 0

CONTSTANT RULE:THE DERIVATIVE OF A CONSTANT FUNCTION IS 0

If f(x)= k, where k is any real number, then f’(x)=0.

Find the Derivative of the following:

57010)(.3

)(.2

10.1

xf

th

y

Find if :dx

dy 3xy

Power Rule

The table to the right gives the function

and its derivative:

nxy

nxy

y y’

x 1

x2 2x

x3 3x2

x4 4x3

x-1 -1 x-2

x1/2 x-1/221

DO YOU NOTICE ANY PATTERNS????

Power Rule

The derivative of is found by multiplying by the exponent n and decreasing the exponent on x by 1

If , then

nxxf )(

1)(' nnxxfnxxf )(

Find the derivative: Power Rule

1.

2.

3.

4.

10)( xxf

2

1)(t

tg

xy

3

5

)( hhp

Constant times a Function

If the derivative exists, the derivative of a constant times a function is the constant times the derivative of the function:

y = k∙g(x)

y’= k∙g’(x)

Example: y = 2x2

Find the Derivative: Constant times a function

1.

2.

3.

4.

5.

6.

47)( xxf

3

2

1)( xxg

85xy

xxf 2)(

3

2

6)( xxf

2

10

ty

Sum or Difference Rule:

The derivative of a sum or difference of a function is the sum or difference of the derivatives:

If f(x) = u(x) ±v(x),

then f ’(x) = u’(x) ± v’(x)

Example: f(x) = x2 + 3x

Find the Derivative: Helpful hint: rewrite before differentiating, if possible

1.

2.

3.

4.

25 23 xxy

x

xxxy

23 2

163

56 23 tttp

2

13)(

xxxf