Search results
Results from the WOW.Com Content Network
The derivative of ′ is the second derivative, denoted as ″ , and the derivative of ″ is the third derivative, denoted as ‴ . By continuing this process, if it exists, the n {\displaystyle n} th derivative is the derivative of the ( n − 1 ) {\displaystyle (n-1)} th derivative or the derivative of order ...
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x 2 y 2. However, a polynomial in variables x and y, is a polynomial in x with coefficients which are polynomials in y, and also a polynomial in y with coefficients which are polynomials in x. The polynomial
For example, antiderivatives of x 2 + 1 have the form 1 / 3 x 3 + x + c. For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ...
For example, the type T of binary trees containing values of type A can be represented as the algebra generated by the transformation 1+A×T 2 →T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed ...
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
2.1 Products. 2.2 Quotients. 2.3 ... or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a ...