3.3, 3.4 Derivatives of composite functions / Derivatives of products and quotients

Derivatives of composite functions / Derivatives of products and quotients, 3.3, 3.4

1)Main Points

Chapter 3.3 introduces a mathematical concept called the chain rule which states that if y=f(z) and z=g(t), then the derivative of the composite function y=f(g(t)) is dy/dx = dy/dz * dz/dt. This in words means that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. This in function form is y=(t+1)^4, or y=z^4 when z=t+1 (where z is the inside function). This is simply a way of splitting up a function into different parts so you can calculate the derivative of a function like y=(4t^2 + 1)^7. 3.4 introduces how to find the derivative of a product or quotient of functions. We notice that simply multiplying the derivatives does not give us the correct answer. We learn that (fg)'= f'g + fg', which means the derivative of a product is the derivative of the first times the second plus the first times the derivative of the second. We find that through the product rule we can create a rule for quotients as well which is: (f/g)'=f'g-fg'/g^2.

2) Challenges

3.3 introduces some challenges for me. I find it really difficult to keep all the things in the chain rule strait. It is easy to forget that at the end of calculating the derivative for the inside function you must multiply it by the derivative of the outside function. I should practice these a little and be able to use them without getting confused. The product/quotient rules make a little bit more sense to me but could also present a lot of logistical problems. It is easy to get the variables confused or getting the basic derivatives wrong could put off the whole problem. It will be important for me to memorize the derivative rules and make sure that I write clearly.

3)Reflections

Both chapters introduced ways of finding the derivatives of more complicated compound functions. This, I'm sure will be useful when we encounter some real world problems that are more difficult to solve than with just simple derivative rules. I hope that I can master these skills well enough to apply them.