#### Answer

$$a)\ outer\ function = \sqrt{x},\ inner\ function = 4x+9x^2$$
$$b)\ outer\ function = tanx,\ inner\ function = x^2 + 1$$
$$c)\ outer\ function = x^5,\ inner\ function = sec(x)$$
$$d)\ outer\ function = x^4,\ inner\ function = 1 + e^x$$

#### Work Step by Step

By using the formula for the chain rule, we know we need to split the function into a composite function that consists of an inner and an outer function. In doing so, we get:
$$a)\ outer\ function = \sqrt{x},\ inner\ function = 4x+9x^2$$
$$b)\ outer\ function = tanx,\ inner\ function = x^2 + 1$$
$$c)\ outer\ function = x^5,\ inner\ function = secx$$
$$d)\ outer\ function = x^4,\ inner\ function = 1 + e^x$$