Before reading this page, make sure you know how to take the derivative of a simple equation like 5x². If you do not, read this page first. Understanding how to take a simple derivative will make the page make much more sense.
When to use the Chain Rule
The chain rule is used of complicated equations, especially when you have some polynomial to the power of something.
Doing the chain rule is done opposite of what you have learned with parenthesis. When doing the chain rule, you have to start with the outside of the parenthesis first. So if you have (5x²+3x+8)³, you will first need to take the derivative of the outside of the parenthesis.
So the first step is to treat 5x²+3x+8 as X. This means your equation would be X³. Taking the derivative of X³ would give you 3x². Now remember X is equal to 5x²+3x+8. So if you put it together, you have 3(5x²+3x+8)².
Next step is to do the inside of the parenthesis. The derivative of 5x²+3x+8 is 10x+3. Now you need to multiply them by each other. So your final product would be 3(5x²+3x+8)²•10x+3. That is the final answer.
Here is another example
. If you have (2x²)³, first you will have 3(2x²)² and 4x. Your answer is 3(2x²)•4x.
More Complicated Ones
When involving Sin(x) or Cos(x), start with the derivative of the trig function. If the trig function is to a power, start with the power, just like earlier in the post.
To do an equation like Sin(5x), you would start by treating 5x as z (or any other variable). Then you would take the derivative of the trig function sin. Taking the derivative of sin gives you cos. Now you need to take the derivative of 5x which gives you 5. Now you multiply them together and it gives you Cos(5x)•5 (usually written at 5Cos(5x)).
More difficult equations such as (sin(5x))³, you would start with the power. So treating sin(5x) as it were Z, you would have Z³. So taking the derivative you would have 3Z². Your Z is actually sin(5x) though, so substitute that back into the equation of 3Z². This gives you 3(sin(5x))². Now you need to take the derivative of inside the parenthesis, which we did one paragraph before this. Then you multiply your answers to give you 3(sin(5x))²•Cos(5x)•5, or 15(sin(5x))²•Cos(5x) when simplified.
When e is involved in an equation, take the original equation multiplied by the derivative of the exponent. So e2x would be just e2x multiplied by the derivative of 2x which is 2. So your answer is just 2e2x.
For more help with other more complicated equations, check out youtube videos. Youtube videos explain more complicated equations much better. Khan Academy does a great job of explaining.
If you have any questions or if you are confused on anything, please let me know. Thanks for reading!