If
y
=
5
x
x
2
+
1
y=5x\sqrt{x^2+1}
y
=
5
x
x
2
+
1
, then
d
y
d
x
\dfrac{dy}{dx}
d
x
d
y
at
x
=
3
x=3
x
=
3
is
5
2
10
\dfrac{5}{2\sqrt{10}}
2
10
5
15
10
\dfrac{15}{\sqrt{10}}
10
15
45
10
+
5
10
\dfrac{45}{\sqrt{10}}+5\sqrt{10}
10
45
+
5
10
45
10
+
15
10
\dfrac{45}{\sqrt{10}}+15\sqrt{10}
10
45
+
15
10
Summary
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Approach
y
=
5
x
x
2
+
1
y=5x\sqrt{x^2+1}
y
=
5
x
x
2
+
1
y
=
5
[
(
x
)
(
x
2
+
1
)
1
2
]
y=5[(x)(x^2+1)^{\frac{1}{2}}]
y
=
5
[(
x
)
(
x
2
+
1
)
2
1
]
d
y
d
x
=
5
[
(
x
)
(
1
2
⋅
1
x
2
+
1
⋅
2
x
)
+
(
x
2
+
1
)
(
1
)
]
\frac{dy}{dx}=5\left[(x)\left(\frac{1}{2}\cdot \frac{1}{\sqrt{x^2+1}}\cdot 2x\right)+(\sqrt{x^2+1})(1)\right]
d
x
d
y
=
5
[
(
x
)
(
2
1
⋅
x
2
+
1
1
⋅
2
x
)
+
(
x
2
+
1
)
(
1
)
]
d
y
d
x
=
5
x
2
x
2
+
1
+
5
x
2
+
1
\frac{dy}{dx} = \frac{5x^2}{\sqrt{x^2+1}}+5\sqrt{x^2+1}
d
x
d
y
=
x
2
+
1
5
x
2
+
5
x
2
+
1
d
y
d
x
∣
x
=
3
=
5
(
3
)
2
3
2
+
1
+
5
3
2
+
1
\frac{dy}{dx}\Big|_{x=3} = \frac{5(3)^2}{\sqrt{3^2+1}}+5\sqrt{3^2+1}
d
x
d
y
∣
∣
x
=
3
=
3
2
+
1
5
(
3
)
2
+
5
3
2
+
1
=
45
10
+
5
10
= \boxed{\frac{45}{\sqrt{10}}+5\sqrt{10}}
=
10
45
+
5
10