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Mathematics
Problem
For
x
>
0
x>0
x
>
0
, let
f
(
x
)
=
cot
(
5
x
2
+
7
)
f(x)=\cot\!\bigl(\sqrt{5x^{2}+7}\bigr)
f
(
x
)
=
cot
(
5
x
2
+
7
)
. Determine
f
′
(
x
)
f'(x)
f
′
(
x
)
. a)
5
x
5
x
2
+
7
csc
2
(
5
x
2
+
7
)
\dfrac{5x}{\sqrt{5x^{2}+7}}\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr)
5
x
2
+
7
5
x
csc
2
(
5
x
2
+
7
)
b)
−
5
x
5
x
2
+
7
sec
2
(
5
x
2
+
7
)
-\dfrac{5x}{\sqrt{5x^{2}+7}}\sec^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr)
−
5
x
2
+
7
5
x
sec
2
(
5
x
2
+
7
)
c)
−
5
x
5
x
2
+
7
csc
2
(
5
x
2
+
7
)
-\dfrac{5x}{\sqrt{5x^{2}+7}}\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr)
−
5
x
2
+
7
5
x
csc
2
(
5
x
2
+
7
)
d)
−
5
x
csc
2
(
5
x
2
+
7
)
-5x\,\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr)
−
5
x
csc
2
(
5
x
2
+
7
)
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