4.1: techniques for differentiation objectives: students will be able to… apply the constant and...
TRANSCRIPT
![Page 1: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/1.jpg)
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
![Page 2: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/2.jpg)
•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
![Page 3: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/3.jpg)
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.
![Page 4: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/4.jpg)
Find the Derivative of the following:
57010)(.3
)(.2
10.1
xf
th
y
![Page 5: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/5.jpg)
Find if :dx
dy 3xy
![Page 6: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/6.jpg)
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????
![Page 7: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/7.jpg)
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 )(
![Page 8: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/8.jpg)
Find the derivative: Power Rule
1.
2.
3.
4.
10)( xxf
2
1)(t
tg
xy
3
5
)( hhp
![Page 9: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/9.jpg)
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
![Page 10: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/10.jpg)
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
![Page 11: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/11.jpg)
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
![Page 12: 4.1: Techniques for Differentiation Objectives: Students will be able to… Apply the constant and power rule for differentiation Apply the sum and difference](https://reader036.vdocuments.us/reader036/viewer/2022082411/5697bfa91a28abf838c99bd9/html5/thumbnails/12.jpg)
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