Problem 1: Evaluate the derivative for the following functions.
Answer
Explanation
We can’t simplify this expression any further, so there’s no hopes for any of our previous methods from Exam 1.
However, since we have a quotient of two functions, we can very easily use the quotient rule to evaluate the derivatives. Recall the quotient rule as the following:
In our case, f(x) will be the function in the numerator...
...and g(x) will be the function in the denominator.
We'll start by evaluating the derivative of f(x), then the derivative of g(x).
Derivative of f(x)
If f(x) equals the following...
...then we can solve for f'(x) with the sum rule...
...and then apply the power rule like so...
...resulting in the following equation for f'(x).
Derivative of g(x)
If g(x) equals the following...
...then we can solve for g'(x) with the sum rule...
...and then apply the power rule like so...
...resulting in the following equation for g'(x).
Combining the functions
To be clear, we now have the following function for f(x), f'(x), g(x), and g'(x):
We'll start by placing f(x) into the quotient rule template...
...then f'(x)...
...then g(x)...
...then g'(x)...
...which results in our final answer!
