3.3 & 3.4 The Chain, Product and Quotient Rules

Main points:
• Chain rule
○ d/dt(f(g(t))=f'(g(t))*g'(t)
○ In words, the derivative of a composite function is the derivative of the outside function times the derivative of the inside function
○ For functions given by formulas the function is first rewritten using a new variable z to represent the inside function
§ y=(t+1)^4 is the same as y=z^4 where z=t+1
○ If z is a differentiable function of t, then
§ d/dt(z^n)=(nz^n-1)(dz/dt)
§ d/dt(e^z)=(e^z)(dz/dt)
§ d/dt(lnz)=(1/z)(dz/dt)
• d/dt(e^kt)=ke^kt
• Product rule
○ d(uv)/dx=(du/dx)*v+u*(dv/dx)
○ In words, the derivative of a product is the derivative of the first times the second, plus the first time the derivative of the second
• Quotient rule
○ (d/dx)(u/v)=[(du/dx)(v) - (u)(dv/dx)] / v^2
○ In words, the derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared

Challenges:
I have studied the chain, product and quotient rule before so I am quite confident with them. In the past I have had trouble with the questions containing lna, a^x and other more complex identities. Therefore I should spend more time on the more challenging problems in order to become comfortable with these mathematical identities.

Reflection:
I was once again looking for examples on economics, and the section in product and quotient rule did have those whereas the section in chain rule did not. I was happy to find several related to the topics such as total revenue and price of a product. I hope doing these examples will prove to be helpful as I am going to take economics on next semester.