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Math 101 Chapter 3: Differentiation
3.29 Equations of tangent and normal
3.31 Higher-order derivatives
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3.30 Summary
In Leibniz’s notation, Leibniz’s rules are:
d
d
x
(
A
f
(
x
)
+
B
g
(
x
)
)
=
A
d
f
d
x
+
B
d
g
d
x
;
{{d}\over{dx}}\Bigl(Af(x)+Bg(x)\Bigr)=A{{df}\over{dx}}+B{{dg}\over{dx}};
d
d
x
(
f
(
x
)
g
(
x
)
)
=
d
f
d
x
g
(
x
)
+
f
(
x
)
d
g
d
x
;
{{d}\over{dx}}\Bigl(f(x)g(x)\Bigr)={{df}\over{dx}}g(x)+f(x){{dg}\over{dx}};
d
d
x
(
f
(
x
)
g
(
x
)
)
=
d
f
d
x
g
(
x
)
-
f
(
x
)
d
g
d
x
g
(
x
)
2
(
g
(
x
)
≠
0
)
.
{{d}\over{dx}}\Bigl({{f(x)}\over{g(x)}}\Bigr)={{{{df}\over{dx}}g(x)-f(x){{dg}% \over{dx}}}\over{g(x)^{2}}}\qquad(g(x)\neq 0).
The chain rule is
d
y
d
x
=
d
y
d
u
d
u
d
x
,
{{dy}\over{dx}}={{dy}\over{du}}{{du}\over{dx}},
the inverse function rule is
d
y
d
x
=
1
/
d
x
d
y
.
{{dy}\over{dx}}=1\bigg/\,\,{{dx}\over{dy}}.