At a Glance - Giving the Correct Answers
Some derivatives are simplify-able, while others aren't. We usually want to simplify the answer a little, but we don't want to do unnecessary work. How do we know when to stop simplifying?
Soon, we'll be spending a lot of energy finding all the places where a particular derivative is zero or undefined. We want to simplify the derivative as much as possible. While we can't give any exact rules for when to stop simplifying, here are some tips that might be useful:
- When using the quotient rule, don't square the denominator. It's easier to see where the denominator is zero if it's in factored form, rather than multiplied out.
- Factor out common factors, it makes the answer tidier, and factoring makes it easier to see where the derivative is 0.
- Cancel things that are common factors in both the numerator and denominator.
- Don't multiply expressions together unless they have only 1 or 2 terms.
These aren't hard and fast rules by any means, but rather general guidelines to help us figure out the best way to work with a messy derivative.
Exercise 1
What's the derivative of the following function?
Exercise 2
What is the derivative of the following function?
Exercise 3
Find the derivative of the function.
Exercise 4
What is f ' (x) for the following function?
Exercise 5
What is f ' (x) for the following function?
- f(x) = e^{x}(x^{2} + 4x)sin x cot x