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Chain Rule
Reference
> Calculus: Differentiation
Description
d
y
d
x
=
d
y
d
u
×
d
u
d
x
\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}
d
x
d
y
=
d
u
d
y
×
d
x
d
u
Examples
d
d
x
sin
(
ln
x
)
\frac{d}{dx} \sin{(\ln{x})}
d
x
d
sin
(
ln
x
)
1
Use
Chain Rule
on
d
d
x
sin
(
ln
x
)
\frac{d}{dx} \sin{(\ln{x})}
d
x
d
sin
(
ln
x
)
. Let
u
=
ln
x
u=\ln{x}
u
=
ln
x
. Use
Trigonometric Differentiation
: the derivative of
sin
u
\sin{u}
sin
u
is
cos
u
\cos{u}
cos
u
.
cos
(
ln
x
)
(
d
d
x
ln
x
)
\cos{(\ln{x})}(\frac{d}{dx} \ln{x})
cos
(
ln
x
)
(
d
x
d
ln
x
)
2
The derivative of
ln
x
\ln{x}
ln
x
is
1
x
\frac{1}{x}
x
1
.
cos
(
ln
x
)
x
\frac{\cos{(\ln{x})}}{x}
x
cos
(
ln
x
)
Done
cos(ln(x))/x
See Also
-
Power Rule
English
Chain Rule