Problem 1: Evaluate the derivative for the following functions.
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 ex + 2x5 + 3x...
...and the external function f(x) being x200 (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!