In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. 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:
a
=
d
v
d
t
=
d
2
x
d
t
2
,
{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}},}
where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression
d
2
x
d
t
2
{\displaystyle {\tfrac {d^{2}{\boldsymbol {x}}}{dt^{2}}}}
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.