Search results
Results from the WOW.Com Content Network
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 slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0. In other words, the value of the constant function, y {\textstyle y} , will not change as the value of x {\textstyle x} increases or decreases.
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 quotient rule states that the derivative of h(x) is
Since the derivative at only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative " ". [ note 2 ] The upshot is that d θ {\displaystyle d\theta } is a one-form on R 2 ∖ { 0 } {\displaystyle \mathbb {R} ^{2}\smallsetminus \{0\}} that is not actually the ...
If there exists an m × n matrix A such that = + ‖ ‖ in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix , and the linear transformation that associates to the increment Δ x ∈ R n the vector A Δ x ∈ R m is, in this general setting ...
2 First derivative identities. ... has curl given by: = where = ±1 or 0 is the ... When the Laplacian is equal to 0, the function is called a harmonic function.
To obtain more general derivative approximation formulas for some function (), let > be a positive number close to zero. The Taylor expansion of f ( x ) {\displaystyle f(x)} about the base point x {\displaystyle x} is
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.