DerivativeIn mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. 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.
Second derivativeIn 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 the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.
Generalizations of the derivativeIn mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. The Fréchet derivative defines the derivative for general normed vector spaces . Briefly, a function , an open subset of , is called Fréchet differentiable at if there exists a bounded linear operator such that Functions are defined as being differentiable in some open neighbourhood of , rather than at individual points, as not doing so tends to lead to many pathological counterexamples.
Derivative testIn calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.
Logarithmic derivativeIn mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f.
Β-Galactosidaseβ-Galactosidase (EC 3.2.1.23, lactase, beta-gal or β-gal; systematic name β-D-galactoside galactohydrolase), is a glycoside hydrolase enzyme that catalyzes hydrolysis of terminal non-reducing β-D-galactose residues in β-D-galactosides. β-Galactosides include carbohydrates containing galactose where the glycosidic bond lies above the galactose molecule. Substrates of different β-galactosidases include ganglioside GM1, lactosylceramides, lactose, and various glycoproteins.
Partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.
Reporter geneIn molecular biology, a reporter gene (often simply reporter) is a gene that researchers attach to a regulatory sequence of another gene of interest in bacteria, cell culture, animals or plants. Such genes are called reporters because the characteristics they confer on organisms expressing them are easily identified and measured, or because they are selectable markers. Reporter genes are often used as an indication of whether a certain gene has been taken up by or expressed in the cell or organism population.
Covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.
Marker geneIn biology, a marker gene may have several meanings. In nuclear biology and molecular biology, a marker gene is a gene used to determine if a nucleic acid sequence has been successfully inserted into an organism's DNA. In particular, there are two sub-types of these marker genes: a selectable marker and a marker for screening. In metagenomics and phylogenetics, a marker gene is an orthologous gene group which can be used to delineate between taxonomic lineages.
