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Mathematics
Mathematics
Problem
Define
y
=
(
x
2
+
4
x
+
7
)
−
3
2
y = (x^{2}+4x+7)^{-\tfrac{3}{2}}
y
=
(
x
2
+
4
x
+
7
)
−
2
3
. (i) Show that
d
y
d
x
=
−
3
2
(
x
2
+
4
x
+
7
)
−
5
2
(
2
x
+
4
)
\displaystyle \frac{dy}{dx} = -\tfrac{3}{2}(x^{2}+4x+7)^{-\tfrac{5}{2}}(2x+4)
d
x
d
y
=
−
2
3
(
x
2
+
4
x
+
7
)
−
2
5
(
2
x
+
4
)
. (ii) Hence prove that
(
x
2
+
4
x
+
7
)
y
′
=
−
3
(
x
+
2
)
y
(x^{2}+4x+7)\,y' = -3(x+2)\,y
(
x
2
+
4
x
+
7
)
y
′
=
−
3
(
x
+
2
)
y
.
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