Derivatives
1. What is a derivative?
2. When looking at a graph of a function, what does the derivative at a specific point represent?
3. What are derivatives useful in determining?
4. Using the derivative rules determine
4a. 5 x 2 {\displaystyle 5x^{2}}
4b.
5
x
2
+
6
x
5
+
17
{\displaystyle 5x^{2}+6x^{5}+17}
4c.
(
3
x
2
+
5
)
(
9
x
7
−
9
)
{\displaystyle (3x^{2}+5)(9x^{7}-9)}
4d.
(
l
n
x
+
7
x
4
+
x
5
)
(
7
x
7
+
2
x
5
−
9
)
{\displaystyle (lnx+7x^{4}+x^{5})(7x^{7}+2x^{5}-9)}
4e.
5
x
2
−
8
x
−
13
x
2
−
5
{\displaystyle {\frac {5x^{2}-8x-13}{x^{2}-5}}}
4f.
7
x
5
−
8
x
5
−
13
x
7
x
3
−
5
{\displaystyle {\frac {7x^{5}-8x^{5}-13x}{7x^{3}-5}}}
4g.
(
7
x
7
+
2
x
5
−
9
)
3
{\displaystyle (7x^{7}+2x^{5}-9)^{3}}
4h.
7
x
7
+
2
x
5
−
9
{\displaystyle {\sqrt {7x^{7}+2x^{5}-9}}}
4i.
s
i
n
(
5
x
)
{\displaystyle sin(5x)}
4j.
9
x
3
−
4
x
2
−
4
x
2
6
x
3
−
10
{\displaystyle {\sqrt {\frac {9x^{3}-4x^{2}-4x^{2}}{6x^{3}-10}}}}
Limits
1. What is a limit?
2. Using limits determine
2a.
lim
x
→
2
2
x
+
1
{\displaystyle \lim _{x\to 2}2x+1}
2b.
lim
x
→
3
5
x
2
−
8
x
−
13
x
2
−
5
{\displaystyle \lim _{x\to 3}{\frac {5x^{2}-8x-13}{x^{2}-5}}}
2c.
lim
x
→
1
x
3
−
1
(
x
−
1
)
2
{\displaystyle \lim _{x\to 1}{\frac {x^{3}-1}{(x-1)^{2}}}}
2d.
lim
x
→
4
3
−
(
x
+
5
)
x
−
4
{\displaystyle \lim _{x\to 4}{\frac {3-{\sqrt {(x+5)}}}{x-4}}}
2e.
lim
x
→
3
x
4
−
81
2
x
2
−
5
x
−
3
{\displaystyle \lim _{x\to 3}{\frac {x^{4}-81}{2x^{2}-5x-3}}}
3. What are the two variations of the definition of the derivative?
4. Using the Definition of the derivative determine
4a. 5 x 2 {\displaystyle 5x^{2}}
4b.
5
x
2
−
3
x
−
7
{\displaystyle 5x^{2}-3x-7}
4c. 4 − x + 3 {\displaystyle 4-{\sqrt {x+3}}}
Implicit Differentiation
1. What makes implicit differentiation problems from standard differentiation problems?
2. Using implicit differentiation methods determine dy/dx for the following
2a. x 3 + y 3 = 7 {\displaystyle x^{3}+y^{3}=7}
2b.
(
x
−
y
)
2
=
x
+
y
+
2
{\displaystyle (x-y)^{2}=x+y+2}
2c.
x
3
y
5
+
x
3
y
2
=
y
{\displaystyle x^{3}y^{5}+x^{3}y^{2}=y}
2d.
x
3
+
y
3
=
7
{\displaystyle x^{3}+y^{3}=7}
2d. e 4 x − e 2 x = y {\displaystyle e^{4x}-e^{2x}=y}
2e.
x
2
=
x
2
+
y
2
{\displaystyle x^{2}={\sqrt {x^{2}+y^{2}}}}
2f.
4
x
2
+
y
3
y
2
−
5
x
{\displaystyle {\frac {4x^{2}+y}{3y^{2}-5x}}}
