Example 1: By Hand

Given y = 3 x 2 9 x + 13 , determine dy dx | x = 3 .

 

Step 1: Find the Derivative

This derivative requires us to apply the power rule.

y = 3 x 2 9 x + 13

 

dy dx = 6 x 9

 

Step 2: Plug the given value into the equation.

In this example we are being asked to plug 3 in for x .

dy dx | x = 3 = 6 ( 3 ) 9

 

dy dx | x = 3 = 9

 

 

Final Result:

The value of the derivative, dy dx , of the function y = 3 x 2 9 x + 13 at x = 3 is 9 .

 

Meaning:

          The instantaneous rate of change of y = 3 x 2 9 x + 13 at x = 3 is 9 .

 

          The slope of y = 3 x 2 9 x + 13 at x = 3 is 9 .

 

 

          The slope of the tangent line to y = 3 x 2 9 x + 13 at x = 3 is 9 .

 

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