Reflection

This week in calculus we learned about the chain rule and began reviewing for first nine week exams. When you solve for the chain rule: you take the derivative of the outside, then derivative of the outside then recopy the inside, multiply by derivative inside.
d/dx f(g(x)) = f'(g(x)) X g'(x)

Examples:

1) y = (4x-1)^3
y = u^2/3 u = 4x-1
3u^2 X 4
3(4x-1)^2 X 4
= 12(4x-1)^2

2) f(t) = (9t+2)^2/3
y = u^2/3 u = 9t+2
2/3u^-1/3 X 9
2/3(9t+2)^-1/3 X 9
6(9t+2)^-1/3
= 6/(9t+2)^-1/3

3) y = sin(pix)^2
cos(pix)^2 X 2pix
= 2pixcos(pix)^2

4) f(x) = cotx/sinx
(sinx)(csc^2x)-[(cotx)(cosx)]/(sinx)^2
sin^2x/sin^3x
1/sinx
= cscx

Also, just to review for the exam, we learned how to do derivative shortcuts, the limit process, discontinuity and much more.
Examples of derivative shortcut:

1) x^3
3x^2-1
= 3x^1

2) 4x^3 +2x -5
4(3)x^3-1 + 2 – 0
= 12x^2 + 2

3) x^7
7x^7-1
= 7x^6