Recall the Chain Rule and how it can be applied to
find the derivative of a composite function. In particular, if is a differentiable
function of , and is a differentiable function of , then
In words, we say that the derivative of a composite function ,
where is considered the "outer" function and the "inner" function, is
"the derivative of the outer function, evaluated at the inner function, times the
derivative of the inner function."
(a) For each of the following functions, use the Chain Rule to find the function's derivative.
Notice the label of each derivative respects the functions name (e.g., the derivative of
should be labeled ).
(i) If , then .
(ii) If , then .
(iii) If , then .
(iv) If , then .
(v) If , then .
(b) For each of the following functions, use your work in (a) to help you determine
the general antiderivative of the function. (Recall that the general antiderivative of a function
includes ``'' to reflect the entire family of functions that share the same derivative.)
Notice the label of each antiderivative respects the name (e.g., the antiderivative of
should be called ). In addition, check your work by computing the derivative of
each proposed antiderivative.
(i) If , then .
(ii) If , then .
(iii) If , then .
(iv) If , then .
(v) If , then .
(c) Based on your experience in parts (a) and (b), conjecture an antiderivative for each
of the following functions. Test your conjectures by computing the derivative of each
proposed antiderivative.
(i) If , then .
(ii) If , then .
(iii) If , then .
You can earn partial credit on this problem.