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In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t).
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
The (unsigned) curvature is maximal for x = – b / 2a , that is at the stationary point (zero derivative) of the function, which is the vertex of the parabola. Consider the parametrization γ(t) = (t, at 2 + bt + c) = (x, y). The first derivative of x is 1, and the second derivative is zero.
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time, [28] and the third derivative is the jerk. [35]
For a function f, if its second derivative f″(x) exists at x 0 and x 0 is an inflection point for f, then f″(x 0) = 0, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third ...
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be