Directorate of distance education

Classical Optimisation
The Derivative
To find a derivative of a function is to differentiate 
with respect to 
a
variable. Properties of derivatives we use in this 
supplement are:

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Notes
219
Where a represents a constant and 
x, y
, and 
z
are variables.
Let’s find the first derivative of the function y = 3
x
2 + 
x 
– 3 with 
respect to the variable 
x
:
In this example, the first derivative of each term is used to find 
the first derivative 
d(y)/dx.
The second derivative is found by taking the 
derivative of the first derivative:
Optimization 
In the calculus, the derivative is taken to find the value of the deci-
sion variable that gives the largest or smallest value of a criterion function. 
The general procedure is to take the first derivative of a function with re-
spect to a decision variable and set the result equal to zero. The equation 
is than solved for the decision variable in terms of the other parameters in 
the equation. To determine whether the optimal point is a maximum, the 
second derivative is taken. If the second derivative is positive, the optimal 
point is a minimum. If the second derivative is negative, the optimal point 
is a maximum. If the second derivative is zero, the point is an inflection 
point.
In the previous example,
The optimal value of 
x
is found by setting this equation equal to 
zero and solving for 
x
:
0 = 6x + 1
x = - (1/6)
When the second derivative was found, it was +6. Therefore
x
= -1/6 is a minimum point.

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Notes
220