Glossary
Constant Multiple Rule
A rule stating that the derivative of a constant multiplied by a function is the constant times the derivative of the function.
Example:
If the cost of producing 'x' items is $10 times the square root of 'x', the Constant Multiple Rule helps you find the marginal cost by keeping the 10 and differentiating the square root of 'x'.
Constant Rule
A rule stating that the derivative of any constant value is always zero.
Example:
If your grade in a class is a fixed 95%, its rate of change (derivative) is 0, illustrating the Constant Rule.
Derivative Rules
Fundamental principles and formulas used to efficiently calculate the derivative of various types of functions, simplifying the process compared to using the limit definition.
Example:
Mastering the Derivative Rules is essential for quickly determining the instantaneous rate of change of complex functions on the AP exam.
Difference Rule
A rule stating that the derivative of a difference between two functions is the difference of their individual derivatives.
Example:
To find how quickly your net savings are changing, you can use the Difference Rule to subtract the derivative of your expenses from the derivative of your income.
Power Rule
A fundamental rule for differentiating functions of the form x^n, where you multiply by the exponent and then reduce the exponent by one.
Example:
To find the rate at which the area of a square changes with respect to its side length (A = s^2), you'd apply the Power Rule to s^2.
Second Derivative
The derivative of the first derivative of a function, often used to determine concavity of a graph or the acceleration of an object.
Example:
If a function describes an object's position over time, its first derivative gives velocity, and its Second Derivative tells you the object's acceleration.
Sum Rule
A rule that allows the derivative of a sum of functions to be found by taking the sum of their individual derivatives.
Example:
When calculating the rate of change of total revenue from two different product lines, you can apply the Sum Rule to differentiate each product's revenue function separately.