Chain Rule

There are rules we can follow to find many derivatives.

• The slope of a constant value (like 3) is always 0
• The slope of a line like 2x is 2, or 3x is 3 etc
• and so on.

If we know the rate of change for two related things, how do we work out the overall rate of change?

The Chain Rule tells us how!

Example: Sage the Dog can run 3 times faster than you, and you can run 2 times faster than me, so Sage can run 3 × 2 = 6 times faster than me.

Let's use some notation. Call the dog "y", me "x" and you can be "u":

dy dx = dy du du dx

• Sage can run 3 times faster than you, so dydu = 3
• You can run 2 times faster than me, so dudx = 2

dy dx = dy du du dx = 3 × 2 = 6

But it is not usually that easy!

Because one function can depend on the current value of the other (which is itself continually changing).

Example: What is d dx sin(x 2 ) ?

There are two functions happening here, sin() and x 2 .

But it is not sin(x), it is sin(the result of x 2 )

Let's use "u" for x 2 so we can have:

dy dx = dy du du dx

d dx sin(x 2 ) = d du sin(u) d dx x 2

The individual derivatives are:

d dx sin(x 2 ) = cos(u) (2x)

Substitute back u = x 2 :

d dx sin(x 2 ) = cos(x 2 ) (2x)

Which is neater this way:

d dx sin(x 2 ) = 2x cos(x 2 )

Notations

There are several different notations that can be used!

Let's do the previous example again using f(g(x)) = f'(g(x))g'(x):

Example: What is d dx sin(x 2 ) ?

sin(x 2 ) is made up of sin() and x 2 :

• f(g) = sin(g)
• g(x) = x 2

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

• f'(g) = cos(g)
• g'(x) = 2x

d dx sin(x 2 ) = cos(g(x)) (2x)

Same result as before (thank goodness!)

Another couple of examples:

Example: What is ddx (1/cos(x)) ?

1/cos(x) is made up of 1/g and cos():

• f(g) = 1/g
• g(x) = cos(x)

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

The individual derivatives are:

• f'(g) = −1/(g 2 )
• g'(x) = −sin(x)

(1/cos(x))’ = −1g(x) 2 (−sin(x))

= sin(x)cos 2 (x)

Note: sin(x)cos 2 (x) is also tan(x)cos(x) or many other forms.

Example: What is ddx (5x−2) 3 ?

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

(5x−2) 3 is made up of g 3 and 5x−2:

• f(g) = g 3
• g(x) = 5x−2

The individual derivatives are:

• f'(g) = 3g 2 (by the Power Rule)
• g'(x) = 5

ddx (5x−2) 3 = (3g(x) 2 )(5)