Considering a composite function , we write a difference quotient for the derivative: . Now let’s define so . Then we substitute that into our difference quotient, giving . We multiply both the numerator and denominator by to get . Clearly from the definition of , so we can consider our limit to be a . Now we notice that the second factor in the limit is a difference quotient that gives , so we now know that .
To find the value of , we use the difference quotient for , but the numerator of that is how we defined (noting our assumption that is differentiable). Therefore and we have . This is the chain rule.
, we write a difference quotient for the derivative: 
. Now let’s define 
so 
. Then we substitute that into our difference quotient, giving 
. We multiply both the numerator and denominator by 
to get 
. Clearly 
from the definition of 
, so we can consider our limit to be a 
. Now we notice that the second factor in the limit is a difference quotient that gives 
, so we now know that 
.
, we use the difference quotient for 
, but the numerator of that is how we defined 
is differentiable). Therefore 
and we have 
. This is the chain rule.