.1.Given two differentiable functions j = j(x) and k = k(x) . By the chain rule, we know

( j k )' (x) = j' (k(x) ) k' (x)

Notice that the right-hand side of this expression is in terms of only j , k , and x (no composition circles ' ' )

(a)Consider three differentiable functions, h = h(x) , j = j(x), and k = k(x) . Write an expression for

( h j k )' (x)

in terms of only h , j , k , and x (no composition circles ' ' )

(b)Consider four differentiable functions, g = g(x) , h = h(x) , j = j(x), and k = k(x) . Write an expression for

( g h j k )' (x)

in terms of only g , h , j , k , and x

(c)Consider five differentiable functions, f = f(x) , g = g(x) , h = h(x) , j = j(x), and k = k(x) . Write an expression for

( f g h j k )' (x)

in terms of only f , g , h , j , k , and x .

(d)Come up with a function that is a composite of five functions. You must use a minimum of three different types of functions. Possible function types include (but are not necessarily limited to) linear, quadratic, cubic, rational, root, exponential, logarithmic, trig, inverse trig, etc. Be creative and original with your composition!

(e)Differentiate the function that you came up with in part (d).