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Often, it is difficult or impossible to solve explicitly for y, and implicit differentiation is the only feasible method of differentiation. An example is the equation =. It is impossible to algebraically express y explicitly as a function of x, and therefore one cannot find dy / dx by explicit differentiation. Using the implicit method ...
Implicit differentiation gives the formula for the slope of the tangent line to this curve to be [3] =. Using either one of the polar representations above, the area of the interior of the loop is found to be 3 a 2 / 2 {\displaystyle 3a^{2}/2} .
Differentiation with respect to time or one of the other variables requires application of the chain rule, [1] since most problems involve several variables. Fundamentally, if a function F {\displaystyle F} is defined such that F = f ( x ) {\displaystyle F=f(x)} , then the derivative of the function F {\displaystyle F} can be taken with respect ...
Suppose that we want to solve the differential equation ′ = (,). The trapezoidal rule is given by the formula + = + ((,) + (+, +)), where = + is the step size. [1]This is an implicit method: the value + appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear.
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
The fifth example shows the possibly complicated geometric structure of an implicit curve. The implicit function theorem describes conditions under which an equation F ( x , y ) = 0 {\displaystyle F(x,y)=0} can be solved implicitly for x and/or y – that is, under which one can validly write x = g ( y ) {\displaystyle x=g(y)} or y = f ( x ...
Implicit differentiation; ... Example 2: Antiderivatives of tangent and cotangent ... Substitution can be used to answer the following important question in ...