Johannes Kepler
1571 – 1630
Johannes Kepler was a German mathematician and astronomer who
discovered that the Earth and planets travel about the sun in elliptical orbits.
He gave three fundamental laws of planetary motion. He also did important
work in optics and geometry.
We now have a pretty good list of “shortcuts” to find
derivatives of simple functions.
Of course, many of the functions that we will encounter are
not so simple. What is needed is a way to combine
derivative rules to evaluate more complicated functions.
Consider a simple composite function:
y  6 x  10
y  2  3x  5 
Let u  3x  5
then y  2u
y  6 x  10
y  2u
u  3x  5
dy
6
dx
dy
2
du
du
3
dx
6  23
dy dy du


dx du dx
and one more, consider:
y  9x  6x 1
2
y   3 x  1
y  9x2  6x  1
y  u2
u  3x  1
dy
 18 x  6
dx
dy
 2u
du
du
3
dx
2
Let u  3x 1
then y  u 2
dy
 2  3 x  1
du
dy
 6x  2
du
This pattern is called
the Chain Rule!
18x  6   6 x  2   3
dy dy du


dx du dx
Chain Rule
dy dy du


dx du dx
If y  f ( g ( x)), let u  g ( x), then y  f (u ).
Differentiate the outside function
evaluate at the inside function…
So
dy dy du
 f (u )  g ( x)  f ( g ( x))  g ( x)


dx du dx
Times the derivative of the
inside function.
If f
f
g is the composite of y  f  u  and u  g  x , then
g  ( x)   f ( g ( x))   f  ( g ( x))  g  ( x)
Chain Rule
We could use substitution:
y  f  g  x    sin  x 2  4 
2
y  sin  x 2  4  let u  x  4
y  sin u
dy
 cos u
du
du
 2x
dx
dy dy du


dx du dx
Chain Rule
dy
 cos u  2 x
dx
dy
 cos  x 2  4   2 x
dx
Here is a faster way to find the derivative:
y  sin  x 2  4 
y  cos  x 2  4  
Differentiate the outside function
evaluate at the inside function…
d 2
x  4

dx
Times the derivative of the
inside function.
Simplifying
y  cos  x 2  4   2 x
Another example:
d
cos 2  3 x 
dx
2
d

cos  3 x  
dx
It looks like we need to use
the chain rule again!
d
 2 cos  3 x    cos  3 x 
dx
derivative of the
outside function
evaluate at the
inside function
derivative of the
inside function
Another example:
d
cos 2  3 x 
dx
2
d

cos  3 x  
dx
d
 2 cos  3 x    cos  3 x 
dx
d
 2 cos  3 x     sin  3 x     3 x 
dx
 2cos  3x   sin  3x   3
 6cos  3x  sin  3x 
The chain rule can be used
more than once.
(That’s what makes the
“chain” in the “chain rule”!)
Examples: Find the derivative of each function.
1.) y  e
2 x3
, find y and y.
2.) f ( x)  1  2 tan x
3.) y  sin 2 (cos(kx)), where k is a constant.
4.) y  sec(3x 2  e x )
5.)
Generalized Derivative formulas include the chain rule!
d n
n 1 du
u  nu
dx
dx
d
du
sin u  cos u
dx
dx
d
du
cos u   sin u
dx
dx
d
du
2
tan u  sec u
dx
dx
and the list continues…
Remember that u is a function of x!
Every derivative problem could be thought of as a chain-rule
problem:
d 2
d
x  2 x  x  2 x 1  2x
dx
dx
derivative of
outside function
evaluated at the
inside function
The derivative of x is one.
derivative of
inside function
It’s all about THE CHAIN RULE!