What is a Derivative?
The 5 Main Derivative Rules
Special Case Derivatives: Your new multiplication tables
1 of 2

Example 1: Power Rule Derivative

Differentiate y = 6 x + 9 with respect to x.

 

Step 1: Simplify and look for algebraic rewrites.

 

Here the equation is a simplified as we need it, and there are no algebra rewrites.

You always have to check though.

y = 6 x + 9

Step 2: Break the problem down into bitesize chunks based upon the + and , and identify the derivative rule for each chunk.

 

Here we have 2 bitesize chunks.

          The first chunk will be, 6x , a power rule .

          The second chunk, 9 , is a constant .

 

When looking at the first chunk we must remind ourselves that there is really an unwritten 1 for the power. That is what allows us to apply the power rule .

 

y = 6 x + 9

 

y = 6 x 1 + 9

Step 3: Apply derivative rules to each of the chunks you identified in Step 2.

 

          We apply the power rule recipe, Bring the power down Subtract 1 from the power, to the first chunk, 6x .

The power is 1 . We bring that 1 down in front and then subtract 1 from that power.

          We apply the constant recipe, the derivative of a constant is always zero, to the second chunk, 9 .

 

To simplify remember that x 0 = 1 .
Anything to the zero power equals 1.

y = 6 x 1 + 9

y = 6 1 x 1 1 + 0

y = 6 x 0 + 0

y = 6 ( 1 ) + 0

y = 6

 

Final Result:

The derivative of y = 6 x + 9  is  y = 6 .

 

Meaning:

          The equation for finding the slope of any tangent line of y = 6 x + 9   is y = 6 .

          The instantaneous rate of change for the entire equation is 6 , there is a constant rate of change.

Which makes sense when we think about the fact that this equation is a line, and a line always has the same rate of change, the same slope, at every x-value .

 

 

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