User:Metallurgia18/sandbox

Partial derivatives
Suppose that f is a function that depends on more than one variable—for instance,
 * $$f(x,y) = x^2 + xy + y^2.$$

f can be reinterpreted as a family of functions of one variable indexed by the other variables:
 * $$f(x,y) = f_x(y) = x^2 + xy + y^2.$$

In other words, every value of x chooses a function, denoted fx, which is a function of one real number. That is,
 * $$x \mapsto f_x,$$
 * $$f_x(y) = x^2 + xy + y^2.$$

Once a value of x is chosen, say a, then f(x, y) determines a function fa that sends y to a2 + ay + y2:
 * $$f_a(y) = a^2 + ay + y^2.$$

In this expression, a is a constant, not a variable, so fa is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies:
 * $$f_a'(y) = a + 2y.$$

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction:
 * $$\frac{\partial f}{\partial y}(x,y) = x + 2y.$$

This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".

In general, the partial derivative of a function f(x1, …, xn) in the direction xi at the point (a1, ..., an) is defined to be:
 * $$\frac{\partial f}{\partial x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_i,\ldots,a_n)}{h}.$$

In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines a function of one variable
 * $$f_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}(x_i) = f(a_1,\ldots,a_{i-1},x_i,a_{i+1},\ldots,a_n),$$

and, by definition,
 * $$\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_i) = \frac{\partial f}{\partial x_i}(a_1,\ldots,a_n).$$

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

This is fundamental for the study of the functions of several real variables. Let $f(x_{1}, ..., x_{n})$ be such a real-valued function. If all partial derivatives $∂f / ∂x_{j}$ of $f$ are defined at the point $a = (a_{1}, ..., a_{n})$, these partial derivatives define the vector
 * $$\nabla f(a_1, \ldots, a_n) = \left(\frac{\partial f}{\partial x_1}(a_1, \ldots, a_n), \ldots, \frac{\partial f}{\partial x_n}(a_1, \ldots, a_n)\right),$$

which is called the gradient of $f$ at $a$. If $f$ is differentiable at every point in some domain, then the gradient is a vector-valued function $∇f$ that maps the point $(a_{1}, ..., a_{n})$ to the vector $∇f(a_{1}, ..., a_{n})$. Consequently, the gradient determines a vector field.