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If

`f`

and

`g`

are the functions whose graphs are shown, let

`u(x)=f(g(x)),v(x)=g(f(x))`

, and

`w(x)=g(g(x))`

. Find each derivative, if it exists. If it does not exist, explain why. (If an answer does not exist, enter DNE.) (a)

`u^(')(1)=`

It does exist.

`u^(')(1)`

does not exist because

`f^(')(1)`

does not exist.

`u^(')(1)`

does not exist because

`g^(')(1)`

does not exist.

`u^(')(1)`

does not exist because

`f^(')(3)`

does not exist.

`u^(')(1)`

does not exist because

`g^(')(2)`

does not exist. (b)

`v^(')(1)=`

It does exist.

`v^(')(1)`

does not exist because

`f^(')(1)`

does not exist.

`V^(')(1)`

does not exist because

`g^(')(1)`

does not exist.

`v^(')(1)`

does not exist because

`f^(')(3)`

does not exist.

`v^(')(1)`

does not exist because

`g^(')(2)`

does not exist. (c)

`w^(')(1)=`

It does exist.

`w^(')(1)`

does not exist because

`f^(')(1)`

does not exist.

`w^(')(1)`

does not exist because

`g^(')(1)`

does not exist.

`w^(')(1)`

does not exist because

`f^(')(3)`

does not exist.

`w^(')(1)`

does not exist because

`g^(')(2)`

does not exist.