Blog for October 23

Overview of the week of 10/17-10/21:

Monday: We reviewed derivatives of exponential and logarithmic functions

Tuesday: PSAT day

Wednesday: NO class

Thursday: Chain rule/derivatives of logarithmic and exponential functions assessment

Friday: Started covering our last type of derivative (!!!), implicit differentiation (derivatives of non-functions)

Implicit differentiation- writing the derivative when y isn't alone, or when the equation can not be solved for y. An example of when implicit differentiation must be used is x2+y2=1

All year we have been finding the derivative of functions in respect to x (without knowing it)

When finding the derivative of an equation like y=x+1 we would find y'=1 or dy/dx= 1, where dy/dx is the derivative of y and 1 is the derivative of x+1.

Now that y is not always alone, and can be on both sides of the equal sign dy/dx has a much bigger role in our derivatives.

Now that we are finding derivatives of y terms that are more than just y (ex. y2, 3y, 17y2) dy/dx is always part of our equations, because it's part of the derivative of y.

When using implicit differentiation it is important to remember to always write dy/dx because it is part of the derivative of the y term

example: The derivative of x=4y is 1= 4(dy/dx)

• 1 is the derivative of x
• 4(dy/dx) is the derivative of 4y

Example from class:

The steps we took when solving this were:

1. Take the derivative
2. Collect dy/dx terms on one side of the = (anything else should be moved to the other side)
3. Factor out dy/dx
4. Divide to get dy/dx alone on one side of the =

**Helpful hint: distributing almost always makes these problems easier

Here is another example from class. This problem involves more than just finding the derivative of an equation: 

Implicit differentiation can also be used to find the second derivative of an equation.

1. Find the first derivative normally, using implicit derivation 
2. Find the second derivative (almost always involves the quotient rule)

Here is an example from class:

**Don't forget: when writing the second derivative dy/dx becomes d2y/dx2