Reflection

This week we reviewed information from chapter 2.
Chain Rule :
you take the derivative of the outside, then derivative of the outside then recopy the inside, multiply by derivative inside.

Derivative of a natural log:

d/dx lnu= 1/u x du

examples:

d/dx ln2x= 1/2x x 2= 1/x

d/dx ln(x^2+1)=1/x^2+1 x 2x= 2x/ x^2+1

d/dx xlnx= x(1/x x 1) +lnx(1)=1+lnx

d/dx lnx^3= 1/x^3 x 3x^2= 3/x

d/dx (lnx)^3= 3(lnx)^2 x 1/x x 1=

3(lnx)^2/x

Implicit functions are not solved for y . Explicit functions are solved for y . When you take the derivative of y you have to put either dy/dx or y' after every y . For x you just take the derivative like normal and constants are always 0 . When solving all you do is take the derivative of the whole equation . After you take the derivative you solve for dy/dx or y ' . And then simplify your answer.

Related Rates
+ When solving related rates problems, you must follow the guidelines.
+ Examples:
1) y = squareroot(x)
1. dy/dt = ? x = 4 dx/dt = 3
2. y = squareroot(x)
3. dy/dt = 1/2x^-1/2
4. dy/dt = 1/2(4)^-1/2
= 3/4
2) xy = 4
1. dy/dt = ? x = 8 dx/dt = 10
2. xy = 4
3. xdy/dt + ydx/dt = 0
4. xdy/dt = -10y
= -10y/x