Search results
Results From The WOW.Com Content Network
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
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
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 primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument , usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their Delta (Δ P ), as is the difference in their function result, the particular notation being determined by the direction ...
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.
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise ...