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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 ).
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
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 ()
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified ...
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.
Applying both sides to e j, the result on each side is the j th partial derivative of f at p. Since p and j were arbitrary, this proves the formula . More generally, for any smooth functions g i and h i on U, we define the differential 1-form α = Σ i g i dh i pointwise by = ()
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.
It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. The fundamental theorem of calculus: If a function f {\displaystyle f} is defined on a partition of the interval [ a , b ] {\displaystyle [a,b]} , b = a + n h {\displaystyle b=a+nh} , and if F {\displaystyle F} is a function whose ...