What is a partial derivative? What does partial derivative mean?

There is a binary function z=f(x,y) and the point (x0,y0) is a point within its definition domain d. Fix y at y0 and let x be partial derivative at x0.

There is an increment x, and accordingly the function z=f(x,y) has an increment (called a partial increment to x) z=f(x0+ x,y0)-f(x0,y0). If the ratio of z to x exists when the limit of x 0 exists, then this limit is called the partial derivative of x of the function z=f(x,y) at (x0,y0). Write as f'x(x0,y0)。

The partial derivative of the function z=f(x,y) in the direction of y is the partial derivative of x at (x0,y0), which is actually the derivative of the unary function z=f(x,y0) at x0 after y is fixed at y0 as a constant Similarly, fix x at x0 so that y has an increment y, if there is a partial derivative at the limit.

Then this limit is called the partial derivative of the function z=(x,y) at (x0,y0) to y. Write as f'y(x0,y0)

The partial derivative of a multivariate function is the one that keeps the other variables constant with respect to the derivative of one of the variables. Find a partial derivative of a variable. Just think of all other variables as constants. For example, f(x,y)=x 2+2xy+y 2

Finding the partial guide for x is f'x=(x^2)'+2y *(x)'=2x+2y

The derivative of a function at a point describes the rate of change of the function around that point. The essence of derivatives is to perform a local linear approximation of a function through the concept of limits. When the independent variable of the missing number f produces an increment h at a point x0, the limit of the increment of the output value of the function to the ratio of the increment of the independent variable h to the value of the increment h when h approaches 0, if it exists, is the derivative of f at x0.

In a univariate function, the derivative is the rate of change of the function. For the study of the "rate of change" of a binary function, the situation is much more complicated because there is one more independent variable.

In the xoy plane, when the moving point changes from p(x0,y0) in different directions, the speed of change of the function f(x,y) is generally different, so it is necessary to study the rate of change of f(x,y) at (x0,y0) in different directions.

The partial derivative of the binary function f to its first independent variable is denoted as f1', the partial derivative of the second independent variable is denoted as f2'The advantage of its fierce bush shed is that it does not need to introduce the symbol of intermediate variables. If the intermediate variables u, v, then f1 are introduced'is the partial derivative of f(u,v) to u, f2'is the partial derivative of f(u,v) to v branches.

f1'with f2'It's still a function of u,v, so it's still a composite function of x,y, and continue to use the derivative of the composite function.

When the function z=f(x,y) is two partial derivatives of (x0,y0).

f'x(x0,y0) and f'When y(x0,y0) are present, we call f(x,y) derivable at (x0,y0). If the function f(x,y) is derivable at every point in the domain d, then the function f(x,y) is said to be derivable in the domain d.

In this case, for each point (x,y) of the domain d, there must be a partial derivative of the slow pair x (for y), thus determining a new binary function in the domain d.

It is called the partial derivative of f(x,y) vs. x (vs. y).

Abbreviated as partial derivative.

According to the definite meaning of the partial derivative, the multivariate function is about an independent variable.

When finding the partial derivative, the rest of the independent variables are treated as constants, and his derivative method is the same as the derivative of the unary function.

The method is the same.

For example, f(x,y)=x 2+2xy+y 2, finding the partial derivative of x is f'x=(x^2)'+2y *(x)'=2x+2y。

Argument. is a binary function of x,y to find the partial derivative of x.

Partial derivation in the x-direction.

There is a binary function z=f(x,y) and the points (x0,y0) are its defining domains.

d One point within. Fix y at y0 and let x have increment x at x0, and correspondingly the function z=f(x,y) has increment (called partial increment to x) z=f(x0+ x,y0)-f(x0,y0).

If the ratio of z to x exists when the limit at x 0 exists, then this limit value is called the partial derivative of the function z=f(x,y) for x at (x0,y0) and is denoted as f'The partial derivative of x(x0,y0) or the function z=f(x,y) at (x0,y0) is actually the bivolt derivative of the number of unary letter Qinghui segments z=f(x,y0) at x0 after y is fixed at y0 as a constant.

Partial derivation in the y direction.

Similarly, fix x at x0 so that y has delta y, and if the limit exists, then this limit is called the partial derivative of the function z=(x,y) at (x0,y0) to y. Write as f'y(x0,y0)。

Geometric meaning and ascension

Represents the tangent slope of a point on a fixed surface.

Partial derivative f'x(x0,y0) represents the tangent slope of a point on the fixed surface to the x-axis; Partial derivative f'y(x0,y0) represents the tangent slope of a point on the fixed surface to the y-axis.

Higher-order partial derivative: If the partial derivative f of the binary function z=f(x,y).'x(x,y) vs. f'y(x,y) is still derivative, then the partial derivatives of these two partial derivatives are called the second-order partial derivatives of z=f(x,y). There are four second-order partial derivatives of the binary jushu function:

f"xx，f"xy，f"yx，f"yy。

Note: f"xy and f"The difference between yx is that the former first finds the partial derivative of x, and then finds the partial derivative of y from the obtained partial derivative function; The latter is to find the partial derivative for y first and then for x.

When f"xy and f"When yx are continuous, the derivative is independent of the order.

In mathematics, a multivariate function is called an old partial derivative, i.e. it keeps the derivatives of one variable constant with respect to the other variables (as opposed to full derivatives, in which all variables are allowed to change). Partial derivatives are useful in vector analysis and differential geometry.

The partial derivative is the derivative of two (four) directions, and the directional derivative can be in any direction, i.e., the partial derivative is the directional derivative of the special dust cleaning.

Partial Derivative:

When the function z=f(x,y) is in (x0,y) the two partial derivatives f of (x0,y0).'x(x0,y0) and f'When y(x0,y0) are present, we call f(x,y) f(x,y) derivable at (x0,y0). If the function f(x,y) is derivable at every point in the domain d, then the function f(x,y) is said to be derivable in the domain d.

At this point, there must be a partial derivative of x (to y) at each point (x,y) corresponding to domain d, so a new binary function is determined in domain d, called the partial derivative of f(x,y) to x (to y). Abbreviated as partial derivative.

According to the definition of partial derivative, when a multivariate function is found as a partial derivative with respect to an independent variable, the rest of the independent variables are regarded as constants, and the derivative method is the same as that of the unary function.

The partial derivative is defined by the limit. Write the limit expression for the partial derivative of a point (x0, y0) according to the definition. At this point, the existence of the limit is consistent with the existence of the partial derivative, so the task of proving the existence of the partial derivative is transformed into proving the existence of the limit.

Extending the data, in order to verify the existence of partial derivatives, such problems usually prove that there is a partial derivative at a certain point. Note that the derivation formula cannot be used at this time.

In the case of unary functions, this is because the derivative function f'(x) calculated by the derivative formula usually contains discontinuity, and f'(x) at discontinuity x0 is coarsely meaningless. For example, fy(x,y) is the partial derivative of y at the point (x,y). It should be noted that here x is treated as a constant.

If you need a partial derivative of y at (0,0), first fix x to x=0, i.e., first find fy(0,y)=[4*(y 3)*e (y 2)] y 2)=4*y*e (y 2), and then replace y=0 to get fy(0,0)=4*0*1=0.

The partial derivative of a multivariate function is its derivative of one variable while keeping the other variables constant (as opposed to the full derivative, allowing all variables to change). Partial derivatives are useful in vector analysis and differential geometry. The partial derivative function is defined as a function of x,y if the partial derivative of z=f(x,y) to x exists at every point (x,y) in the d region, and is called the partial derivative of the rock-land function z=f(x,y) to the independent variable x.

Similarly, for the partial derivative function of y, it should be noted that the partial derivative function can be biased not only at a certain point, but also on d in a certain region. If z=f(x,y) has a partial derivative at p(x,y), then the point p must belong to region d, i.e., region d. Caution or therefore, we can naturally assume that a domain of point p belongs to the region d, and therefore there must also be a partial derivative function in a domain of point p.