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The graph of =, with a straight line that is tangent to (,). The slope of the tangent line is equal to . (The axes of the graph do not use a 1:1 scale.) The derivative of a function is then simply the slope of this tangent line.
This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be ...
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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.
In SI, this slope or derivative is expressed in the units of meters per second per second (/, usually termed "meters per second-squared"). Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the ...
The higher-order derivatives are less common than the first three; [1] [2] thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics. [ 3 ] The fourth derivative is referred to as snap , leading the fifth and sixth derivatives to be "sometimes somewhat facetiously" [ 4 ] called crackle ...
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
Both of these classical methods have problems with calculating higher derivatives, where complexity and errors increase. Finally, both of these classical methods are slow at computing partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Automatic differentiation solves all of ...