Derivatives

A two variable equation can be graphed on a simple two dimensional grid.  It will often be some sort of curved line.  The derivative is an equation that gives the slope of that line at a given point.

Example Equation

Can also be expressed as

Derivative

How was it derived?

The following power rule is applied to each term:

The way I think of it is that the “t” in the power rule is the entire term, which is usually a constant, a variable, and an exponent.

First term

The first term of the example equation is

Which when thought of as the “t” in the power rule with constant, variable, and exponent looks like this:

Applying the power rule means taking the exponent as the value “n”.  In this case 4.  Multiplying it by the front of the term is (4 x 1) or 4.  Changing the exponent to n-1 is (4-1) or 3.  So the first term derives to:

Second Term

Looking at the second term of the example equation and what it derives into shows the 3 being multiplied by the 2 and the exponent decreasing to 2.

Third Term

The third term is similar and becomes 8x.  The exponent was reduced to 1 so it need not be displayed anymore.

Fourth Term

The fourth term sees the exponent reducing down to zero, and since anything to the zeroth power is one, the x to the zeroth power becomes 1 and doesn’t need to be displayed anymore.  It just multiplies by the 5 and becomes the value 5.

Fifth Term

The fifth term is just a constant so like all constants, it derives to zero, so is effectively just dropped.  Why?  Because to get it into the “t” form for the power rule, it becomes:

With the starting exponent of zero, that means “n” is zero, and since the whole thing is going to be multiplied by “n” on the front, the whole term solves to zero.