Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives of z and y. More precisely, if
h
=
z
∘
y
{\displaystyle h=z\circ y}
is the composition such that
h
(
x
)
=
z
(
y
(
x
)
)
{\displaystyle h(x)=z(y(x))}
for every x, then the chain rule is, in Lagrange's notation,
h
′
(
x
)
=
z
′
(
y
(
x
)
)
y
′
(
x
)
.
{\displaystyle h'(x)=z'(y(x))y'(x).}
or, equivalently,
h
′
=
(
z
∘
y
)
′
=
(
z
′
∘
y
)
⋅
y
′
.
{\displaystyle h'=(z\circ y)'=(z'\circ y)\cdot y'.}
The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y. In this case, the chain rule is expressed as
d
z
d
x
=
d
z
d
y
⋅
d
y
d
x
,
{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}
and
d
z
d
x
|
x
=
d
z
d
y
|
y
(
x
)
⋅
d
y
d
x
|
x
,
{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}
for indicating at which points the derivatives have to be evaluated.
In integration, the counterpart to the chain rule is the substitution rule.
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