Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if u is a differentiable function of x, and f is a differentiable function of "(x), then d . 1 E [f(u(x))] = f ("(30) ' u (x)- In words, we say that the derivative of a composite function C(x) = f(u(x)), where f is considered the "outer" function and u 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 nd the function's derivative. Be sure to label each derivative by name (e.g., the derivative of 9(x) should be labeled g'(x)). 1- RU") = 331 iv. q(x) : (2 7x)4 ii. h(x) = sin(5x + 1) v. r(x) : 34-1" iii. p(x) = arctan(2x) b. For each of the following functions, use your work in (a) to help you determine the general antiderivative' of the function. Label each anderivative by name (e.g., the antiderivative of 111 should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. i. m(x) : 83-" iv. v(x) : (2 7x)3 ii. n(x) = cos(5x +1) v. 10(35): 3411;- ... - 'l 111. 5(x) _ ""2 c. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following func- tions. Test your conjectures by computing the derivative of each proposed antiderivative. i. a(x) = cos(nx) iii. C(x) = max2 ii. b(x) = (4x + 7)11