Seven Swans a Swimming Christmas Post

Where the Derivative Doesn't Exist
There are four functions/graphs where the derivative doesn't exist.

One of these is the absolute value function. The derivative doesn't exist at the corner.
f(x) = |x|

Another graph where the derivative doesn't exist is a cusp.

Piece-wise functions where there is a jump in the graph also don't have derivatives at the jump.

Rational functions lack a derivative at the asymptote.


Implicit Differentiation
Implicit differentiation is where you take the derivative of a function and you have multiple variables. When you find the derivative of each piece, you must multiply it by d(variable)/d(variable you are finding with regards to). Here is an example:

y² + x² = 3 Find dy/dx
2y(dy/dx) + 2x(dx/dx) = 0
2y(dy/dx) + 2x = 0
2y(dy/dx) = -2x
dy/dx = -x/y


The most important thing to remember while solving related rates problems is multiplying each piece by the correct d(variable)/d(variable). Otherwise, you will most likely not get the correct answer.