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In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f ( x ) {\displaystyle y=f(x)} can be denoted by
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise ...
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
An example of the use of calculus in mechanics is Newton's second law of motion, which states that the derivative of an object's momentum concerning time equals the net force upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times its acceleration , which is the time derivative of ...
In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
The simplest example is the ring of dual numbers R[ε], where ε 2 = 0. This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p) belongs to the ideal I p of functions on R which vanish at p.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
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