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The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
A coefficient is usually a constant quantity, but the differential coefficient of f is a constant function only if f is a linear function. When f is not linear, its differential coefficient is a function, call it f ′, derived by the differentiation of f, hence, the modern term, derivative. The older usage is now rarely seen.
When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
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In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. 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.
That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X. The exterior product of differential forms (denoted with the same symbol ∧ ) is defined as their pointwise exterior product .
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The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case, ∂ f ∂ x = y . {\displaystyle {\frac {\partial f}{\partial x}}=y.} However, if y depends on x , the partial derivative does not give the true rate of change of f as x changes because the partial derivative assumes that y is fixed.