find the derivative of√x(√x+1)
Answers
Answer:
Step-by-step explanation:
The details depend on whether you use
f
'
(
a
)
=
lim
x
→
a
f
(
x
)
−
f
(
a
)
x
−
a
or,
f
'
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
.
For
f
(
x
)
=
√
x
+
1
You will need to evaluate one of:
f
'
(
a
)
=
lim
x
→
a
√
x
+
1
−
√
a
+
1
x
−
a
or,
f
'
(
x
)
=
lim
h
→
0
√
x
+
1
+
h
−
√
x
+
1
h
.
Using either from of the definition, you'll use:
(
√
u
−
√
v
)
(
√
u
+
√
v
)
=
u
−
v
For the first form of the definition, we get:
f
'
(
a
)
=
lim
x
→
a
√
x
+
1
−
√
a
+
1
x
−
a
=
lim
x
→
a
(
√
x
+
1
−
√
a
+
1
)
(
x
−
a
)
(
√
x
+
1
+
√
a
+
1
)
(
√
x
+
1
+
√
a
+
1
)
=
lim
x
→
a
(
x
+
1
)
−
(
a
+
1
)
(
x
−
a
)
(
√
x
+
1
+
√
a
+
1
)
=
lim
x
→
a
1
√
x
+
1
+
√
a
+
1
=
1
2
√
a
+
1
Using the second form of the definition is similar, but you'll end up with:
f
'
(
x
)
=
lim
h
→
0
√
x
+
1
+
h
−
√
x
+
1
h
=
lim
h
→
0
(
x
+
1
+
h
)
−
(
x
+
1
)
h
(
√
x
+
1
+
h
+
√
x
+
1
)
=
lim
h
→
0
h
h
(
√
x
+
1
+
h
+
√
x
+
1
)
=
1
2
√
x
+
1