In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression is the second derivative of position (x) with respect to time. On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way. Second derivative power rule The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows: Notation The second derivative of a function is usually denoted . That is: When using Leibniz's notation for derivatives, the second derivative of a dependent variable with respect to an independent variable is written This notation is derived from the following formula: Alternative notation As the previous section notes, the standard Leibniz notation for the second derivative is . However, this form is not algebraically manipulable. That is, although it is formed looking like a fraction of differentials, the fraction cannot be split apart into pieces, the terms cannot be cancelled, etc. However, this limitation can be remedied by using an alternative formula for the second derivative. This one is derived from applying the quotient rule to the first derivative. Doing this yields the formula: In this formula, represents the differential operator applied to , i.e., , represents applying the differential operator twice, i.e., , and refers to the square of the differential operator applied to , i.e., . When written this way (and taking into account the meaning of the notation given above), the terms of the second derivative can be freely manipulated as any other algebraic term. For instance, the inverse function formula for the second derivative can be deduced from algebraic manipulations of the above formula, as well as the chain rule for