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Here we compute derivatives of compositions of functions
So far we have seen how to compute the derivative of a function built up from other
functions by addition, subtraction, multiplication and division. There is another
very important way that we combine functions: composition. The chain rule
allows us to deal with this case. Consider While there are several different
ways to differentiate this function, if we let and , then we can express . The
question is, can we compute the derivative of a composition of functions
using the derivatives of the constituents and ? To do so, we need the chain
Chain Rule If and are differentiable, then
It will take a bit of practice to make the use of the chain rule come naturally, it is
more complicated than the earlier differentiation rules we have seen. Let’s return to
our motivating example.
Set and , now Hence
Let’s see a more complicated chain of compositions.
Set and . Hence, and by the chain rule we know Since We have that
The chain rule allows to differentiate compositions of functions that would otherwise
be difficult to get our hands on.
Set , ,and so that . Now
Using the chain rule, the power rule, and the product rule it is possible to avoid using
the quotient rule entirely.
Rewriting this as set and so that . Now and by the product and chain rules Since
and , write
Now that we are getting comfortable with chain rule, try one other.
Suppose is a
function whose values are given in the following table.
At it’s outer most level, this is a product of and , so we start with product rule.