Partial derivative

In 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. In 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 f ( x , y , … ) {displaystyle f(x,y,dots )} with respect to the variable x {displaystyle x} is variously denoted by Sometimes, for z = f ( x , y , … ) , {displaystyle z=f(x,y,ldots ),} the partial derivative of z {displaystyle z} with respect to x {displaystyle x} is denoted as ∂ z ∂ x . {displaystyle { frac {partial z}{partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), though he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol again in 1841. Suppose that f is a function of more than one variable. For instance, The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the x z {displaystyle xz} -plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P ( 1 , 1 ) {displaystyle P(1,1)} and parallel to the x z {displaystyle xz} -plane, we treat y {displaystyle y} as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane y = 1 {displaystyle y=1} . By finding the derivative of the equation while assuming that y {displaystyle y} is a constant, we find that the slope of f {displaystyle f} at the point ( x , y ) {displaystyle (x,y)} is: So at ( 1 , 1 ) {displaystyle (1,1)} , by substitution, the slope is 3. Therefore, at the point ( 1 , 1 ) {displaystyle (1,1)} . That is, the partial derivative of z {displaystyle z} with respect to x {displaystyle x} at ( 1 , 1 ) {displaystyle (1,1)} is 3, as shown in the graph.