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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
When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several
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 derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines. Here the function ...
The red curve is the graph of function with 3 roots in the interval [−3, 2]. Thus its second derivative (graphed in green) also has a root in the same interval. The requirements concerning the n th derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand ...
If y = f(x 1, ..., x n) and all of the variables x 1, ..., x n depend on another variable t, then by the chain rule for partial derivatives, one has = = + + = + +. Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantity dt.
If f is a function, then its derivative evaluated at x is written ′ (). It first appeared in print in 1749. [3] Higher derivatives are indicated using additional prime marks, as in ″ for the second derivative and ‴ for the third derivative. The use of repeated prime marks eventually becomes unwieldy.