It would be better to rephrase the definitions of derivative and integral functions as shown below to make them more technically accurate.

Current

$integral

Returns an approximation for the mathematical integral value, which is calculated using the trapezoidal rule with each set of adjacent documents forming a trapezoid using the:

• sortBy field values in the $setWindowFields stage for the x axis values.
• input field expression result values in $integral for the y axis values.

Updated

$integral

Returns the approximation of the area under the curve calculated using the trapezoidal rule where each set of adjacent documents form a trapezoid using the:

• sortBy field values in the $setWindowFields stage for the integration intervals.
• input field expression result values in $integral for the y axis values.

Current

$derivative

Returns the mathematical derivative value, which is calculated using the:

• First and last documents in the $setWindowFields stage window.
• Numerator, which is set to the result of subtracting the numeric expression value for the first document from the expression value for the last document.
• Denominator, which is set to the result of subtracting the sortBy field value for the first document from the sortBy field value for the last document.

Updated

$derivative

Returns the average rate of change within the specified window, which is calculated using the:

• First and last documents in the $setWindowFields stage window.
• Numerator, which is set to the result of subtracting the numeric expression value for the first document from the expression value for the last document.
• Denominator, which is set to the result of subtracting the sortBy field value for the first document from the sortBy field value for the last document.