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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.
for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
In Newton's notation or the dot notation, a dot is placed over a symbol to represent a time derivative. If is a function of , then the first and second derivatives can be written as ˙ and ¨ , respectively.
The function has second derivative ; thus it is convex on the set where and concave on the set where Examples of functions that are monotonically increasing but not convex include f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} and g ( x ) = log x {\displaystyle g(x)=\log x} .
However, Leibniz did use his d notation as we would today use operators, namely he would write a second derivative as ddy and a third derivative as dddy. In 1695 Leibniz started to write d 2 ⋅x and d 3 ⋅x for ddx and dddx respectively, but l'Hôpital, in his textbook on calculus written around the same time, used Leibniz's original forms. [18]
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 = = =.
The mean value theorem gives a relationship between values of the derivative and values of the original function. If f ( x ) is a real-valued function and a and b are numbers with a < b , then the mean value theorem says that under mild hypotheses, the slope between the two points ( a , f ( a )) and ( b , f ( b )) is equal to the slope of the ...
The second derivative of f is the everywhere-continuous 6x, and at x = 0, f″ = 0, and the sign changes about this point. So x = 0 is a point of inflection. More generally, the stationary points of a real valued function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } are those points x 0 where the derivative in every ...