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The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h).: 237 The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.
This, combined with the sum rule for derivatives, shows that differentiation is linear. The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is if it is differentiable.)
Differentiation is linear. The product rule. The chain rule. The inverse function rule. Power laws, polynomials, quotients, and reciprocals. The polynomial or elementary power rule. The reciprocal rule. The quotient rule. Generalized power rule.
The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken.
A sum-product number in a given number base is a natural number that is equal to the product of the sum of its digits and the product of its digits. There are a finite number of sum-product numbers in any given base . In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS ): 0, 1, 135, and 144.
The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then ( pq ) ∗ = q ∗ p ∗ , not p ∗ q ∗ . The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det (A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero ...
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of ...