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Let the function z = x / y. find the total differential dz| (2,1)
DZ / DX = 1 / y, the value at (2,1) is 1
DZ / dy = - X / y ^ 2, the value in (2,1) is - 2
So dz| (2,1) = DX - 2dy
Let the function z = x square y + 1, find the total differential DZ
z=x ² y+1
∂z/∂x=2yx
∂z/∂y=x ²
So DZ = 2yxdx + X ² dy
22. It is known that the binary implicit function z = Z (x, y) is determined by the equation Z ^ 2 + YZ = 1-xsiny, and the total differential DZ is obtained
2zdz+zdy+ydz=-sinydx-xcosydy
dz=[-sinydx-(xcosy+z)dy]/(2z+y)
4. The binary implicit function z = Z (x, y) is determined by the equation sec Z + YZ ^ 2 = 1-x ^ 3Y, and the total differential DZ is obtained
SEC Z + YZ ^ 2 = 1-x ^ 3Y derivative of X: secztanz * z'x + 2yzz'x = - 3x ^ 2Y z'x = - 3x ^ 2Y / (secztanz + 2yz) derivative of Y: secztanz * z'y + Z ^ 2 + 2yzz'y = - x ^ 3 z'y = (- x ^ 3-z ^ 2) / (secztanz + 2yz) DZ = [- 3x ^ 2Y / (secztanz + 2yz)] DX + [(- x ^ 3-z ^ 2) / (secztanz +
What is the difference between differential and derivative of binary function?
Differential generally refers to total differential or total derivative. There is no difference in this respect. If it is partial derivative, there is a difference
For example, u = x ^ 2Y
His total differential or total derivative is generally written as: Du = 2ydx + x ^ 2dy
But partial derivative to x = 2Y, partial derivative to y = x ^ 2
What is the relationship between the derivative of a function at a point and the differential of a variable at that point What does it have to do with the differential of the function
① For a univariate function y = f (x), there is no difference between derivative and differential
The geometric meaning of derivative is the instantaneous change rate of curve y = f (x), i.e. tangent slope
Differential refers to the ratio △ y = △ f (x + △ x) - f (x) between the increment of the dependent variable and the increment of the independent variable. Here, the independent variable x can be regarded as a function y = x about itself,
So △ x = △ y, so another way of saying differential is called derivative. Dy / DX is the ratio of two variables. Generally speaking, dy / DX = y '
② For multivariate functions, such as bivariate function z = f (x, y), the derivative becomes a partial derivative about a variable. At this time, the differential symbol DZ / DX is a whole and cannot be understood apart. Moreover, there is an important difference that the derivative is not necessarily differentiable. That is, the derivative is a necessary and insufficient condition for differentiability
However, there is a theorem that if the partial derivative is continuous, the function is differentiable. See the chapters on total differential and partial derivative for details
RELATED INFORMATIONS
- 1. Is the total differential of a binary function the partial derivative?
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