Computing Derivatives - Solving Derivatives

Solving Derivatives

There are a lot of rules floating about now. Besides knowing how to take the derivatives of less complicated functions, we have all these rules for taking the derivatives of more complicated functions:

These rules can be combined in all sorts of ways. How do we know which one(s) to use?

There are several parts to the answer,

• Practice. Every derivative calculated helps our sense of what we should be doing to find that next derivative.
    
• Keep track of the work carefully, like we did when finding the derivative of a product of 3 functions.
    
• Rewrite functions before finding derivatives.

• Find derivatives the simplest way - for example, use the multiplication-by-a-constant rule instead of the quotient rule
  to find the derivative of .
    
• Work from the outside in. We haven't talked about this yet, but we will.

• Using the Correct Rule(s)

  "Work from the outside in" is a suggestion for how to organize our work when the derivatives become more difficult to manage. We want to start with the outer-most operation, and work in from there. The "outer-most" operation is the one that would be performed last if we were typing things into a calculator. This is similar to top-down design in computer programming.

• 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.