Method: Derivatives Inside Derivatives

These are the steps that I go through on every derivative I am trying to find.

Step 1: Simplify and look for algebraic rewrites.

This is your new Option 1. This means it is the first thing you will always want to do before you start actually applying a derivative rule. You must ensure that the equation is ready to have a derivative taken.

• Roots or Radicals
• x’s on the bottom of a fraction.
• Trig functions raised to a power.

Step 2: Identify your primary derivative rule.

Step 3: Break the problem up into its bite-size chunks based on the primary derivative rule you chose in Step 2.

All of our main derivative rules and special cases involve breaking up a larger problem into its bite-size chunks. It is these bite-size chunks where we will find our secondary derivative rules we will need to apply.

Step 4: Identify and apply any secondary derivative rule you need to find the derivative of each of your bite-size chunks.

Keep in mind that derivative rules are really just a way to take an equation that is big and nasty, and break it down into little bite-size math problems that are able to be digested. This means that you need to now look at your little bite-size chunks and apply the same method to decide what derivative rule you need to apply.

You basically treat each one of your bite-size chunks as its own math problem, and ask yourself what derivative rule you would need to apply to find the derivative of that bite-size chunk. Essentially play the eye squinting game again for each of your bite-size chunks.

This may lead you to even further secondary derivative rule, and you may have to repeat this step several times.

Step 5: Bring it all back together following the primary derivative rule.