Search results
Results from the WOW.Com Content Network
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 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 ).
Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.
Another common notation for differentiation is by using the prime mark in the symbol of a function . This is known as prime notation , due to Joseph-Louis Lagrange . [ 22 ] The first derivative is written as f ′ ( x ) {\displaystyle f'(x)} , read as " f {\displaystyle f} prime of x {\displaystyle x} , or y ...
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 ...
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin ′ ( a ) = cos( a ), meaning that the rate of change of sin( x ) at a particular angle x = a is given ...
The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point.
Suppose we want to differentiate () = (). By using the product rule, one gets the derivative f ′ ( x ) = 2 x ⋅ sin ( x ) + x 2 cos ( x ) {\displaystyle f'(x)=2x\cdot {\text{sin}}(x)+x^{2}{\text{cos}}(x)} (since the derivative of x 2 {\displaystyle x^{2}} is 2 x , {\displaystyle 2x,} and the derivative of the sine function is the cosine ...