Instruction
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Differential (from lat. "the difference") is the linear part of the full increment function. Differential denoted by df, where f is a function. The function of a single argument is sometimes portrayed dxf or dxF. Suppose there is a function z = f(x, y), a function of two arguments x and y. Then the total increment of the function will be:
f(x, y) – f(x_0, y_0) = f _x (x, y)*(x – x_0) + f _y(x ,y)*(y – y_0) + α, where α is infinitesimal (α → 0), which is ignored in the definition of the derivative as lim α = 0.
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Differential of function f for argument x is a linear function of relative increment (x – x_0), i.e. df(x_0) = f _x_0 (Δx).
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Geometrical meaning of the differential of a function: if a function f is differentiable at x_0, then its differential at this point is the increment of the ordinate (y) a tangent line to the graph of the function.

The geometric meaning of the total differential of a function of two arguments is three – dimensional analogue of the geometric meaning of the differential of a function of one argument, i.e. that the increment of applicati (z) the tangent plane to the surface whose equation is set to a differentiable function.
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You can write the total differential of a function using the increment function and arguments is a more common form of entry:
Δz = (δz/δx)dx + (δz/δy)dy, where δz/δx is the derivative of the function z for the argument x, δz/δy is the derivative of the function z for the y argument.
Say that a function f(x, y) differentiable at (x, y), if such x and y values you can define the total differential of this function.

The expression (δz/δx)dx + (δz/δy)dy is the linear part of the increment of the original function, where (δz/δx)dx is the differential of a function z on x, and (δz/δy)dy the differential of y. While differentiation according to one of the arguments assume that the other argument or arguments (if several) - constant values.
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Example.
Find the total differential of the following function: z = 7*x^2 + 12*y - 5*x^2*y^2.

Solution.
Using the assumption that y is a constant, find the partial derivative in the argument x,
δz/δx = (7*x^2 + 12*y - 5*x^2*y^2)’dx = 7*2*x + 0 – 5*2*x*y^2 = 14*x – 10*x*y^2;
Using the assumption that x is a constant, find the partial derivative for the y argument:
δz/δy = (7*x^2 + 12*y - 5*x^2*y^2)’dy = 0 + 12 – 5*2*x^2*y = 12 – 10x^2*y.
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Write down the total differential of a function:
dz = (δz/δx)dx + (δz/δy)dy = (14*x – 10*x*y^2)dx + (12 – 10x^2*y).