3.1 Power Rule - Calculus
๏ปฟ3.1 Power Rule
Write your questions here!
Notation
()
POWER RULE
=
Find the derivative of the following.
()
=
37
-
45
-
1 3
3
+
2
-
3
+
7
NOTES
= 2-3 + 4 +
Rewrite and then take the derivative.
= 37 - + 252
()
=
1
+
4 2
-
1 (3)2
()
=
-162+5-1 2
Evaluate
()
=
1 2
4
-
4-2
+
Find (3)
Find =4
1 = + 4
Higher Order Derivatives
() = 7 - 24 + 52 - 3 + 9 () = () = () = (4)() =
= + -2 = 2 2 =
Find Derivative on the Calculator
()
=
1 2
2
-
1
(4) =
() =
() = 1 + csc () = 2 =
Derivative means...
Slope at a point
Given
=
1 2
4
-
+
2
find
the slope at = 2
Slope of the tangent line
Write the equation of the line
tangent to
=
1 2
4
-
+
2
at = 2
Instantaneous rate of change
What is the instantaneous rate of change at 3 seconds? () = -4.92 + 40 + 6
Normal Line
Write the equation of the normal line at = 3 and then graph it! () = 3 - 42 + + 3
SUMMARY:
Derivative Rules
Constant Rule = 0
Power Rule
=
-1
Now, summarize your notes
here!
Constant Multiple Rule () =
Sum/Difference Rule
( ? ) = ?
3.1 Power Rule
Find the derivative of the following.
1. () = 23 - 4 + 5
2. = 3100 - 28 - 7
PRACTICE
3.
()
=
5-2
-
1 2
4
4.
()
=
6 3
+
62/3
-
41/2
+
2
5.
()
=
1 3
+
12
6.
=
3 -2
-
1 (6)2
7. () = + 33 + 2
8. = 32 + 847
9.
()
=
1
+
3 6
10.
()
=
1
+
3 52
13.
()
=
4 3
3
11. () = -162 + 40 + 5
12. = 2 -
14.
()
=
23+4-5
15.
()
=
63+42-9 3
Find the derivatives of the following.
16. () = 37 - 43 + 5 + 7
17. = 4 +
() = () =
=
() = (4)() =
2 2 =
18.
=
1 3
-
1 2
4
+
2
=
=
=
Given () = - + ,
()
=
+
,
and
() = , find the following.
19. (2) =
20. (-3) =
21. 2(4) =
22. Find the slope of () at = 3. 23. At what value of x is () = 0 ? 24. What is the slope of the tangent line of () at the point (16, 4) ?
Find the equation for the slope of the line tangent to the given function.
25. () = 2 - 2
26.
=
-23
+
1 2
2
-
7
+
5
27.
()
=
1 2
-
1 2
Is the slope of the tangent line positive, negative, or zero at the given point?
28.
()
=
43-162 2
at = 2
29. = 24 + 53 at = -2
30. () = 335 - 4-1 at = 8
Write the equation of the tangent line and the normal line at the point given.
31. () = 3 + 4 at = 4
32.
=
2+3-4 2
at = 8
The function is graphed below. Write the equation of the tangent line at the given point and graph it.
33. () = -2 - 2 - 1 at = -2
34.
=
3 2
-
2 2
-
at = 1
The function is graphed below. Write the equation of the normal line at the given point and graph it.
35. () = -3 + 22 - 2 at = 2
36. = 2 + 6 + 9 at = -2
You will need to use a graphing calculator for 37-42
Use the graph to find the derivative of the function at the given value. Round to nearest thousandth.
37.
()
=
2+1 -2
at
=6
38. = at = -1
39.
()
=
2
sin
at
=
2
Write the equation of the tangent line at the point given and sketch the graph. Round to nearest thousandth.
40. () = -3 + 4 at = 5
41. = ln() + 4 at =
42.
()
=
csc
+
1
at
=
4
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