**Problem 1:** Evaluate the derivative for the following functions.

## Answer

## Explanation

This problem cannot be simplified using Exam 1 techniques. However, we can see we have a composite function present, with the internal function g(x) being e^{x} + 2x^{5} + 3x...

...and the external function f(x) being x^{200} (with x essentially representing the internal function g(x)).

Therefore, since we're dealing with a composite function here, we'll utilize the chain rule, which has the following template:

Let's start by solving for f'(g(x)).

### Solving for f'(g(x))

To start, let's recall that f(x) in our composite function equals the following:

Now, let's evaluate for the derivative f'(x).

Okay great... but how do we get from f'(x) to f'(g(x))?

All we've gotta do is plug in g(x) for x like so!

### Solving for g'(x)

We found g(x) to be the following:

To evaluate for g'(x), we'll utilize the sum rule...

...which when simplified...

...results in the following derivative function for g'(x)!

### Combining the functions

Now, let's put these together in the template formula for the composite rule.

We'll start by placing f'(g(x)) like so...

...and then g'(x) like so:

And with that, we have our final answer!

I’m a Miami University (OH) 2021 alumni who majored in Information Systems. At Miami, I tutored students in Python, SQL, JavaScript, and HTML for 2+ years. I’m a huge fantasy football fan, Marvel nerd, and love hanging out with my friends here in Chicago where I currently reside.