# Partial Derivatives

## Definition

Partial derivatives: definition

Assume that $$f(x,y)$$ is a function of two variables. The partial derivative $$f_x$$ of $$f$$ with respect to $$x$$ at the point $$(a,b)$$ is defined as:

$f_x(a,b)=\lim_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}.$

Similarly, $$f_y(a,b)$$ is defined as:

$f_y(a,b)=\lim_{h\to 0}\frac{f(a,b+h)-f(a,b)}{h}.$

Sometimes we use notation $$\frac{\partial }{\partial x}f$$, $$\frac{\partial f}{\partial x}$$, $$\frac{\partial }{\partial x}f(x,y)$$, or $$\partial_x f(x,y)$$ for partial derivatives.

## Explanation

Consider the following function of two variables:

$g(x,y)=\cos(x^3+y^2).$

Using this function we can make single-variable functions: $$\psi(x)=g(x,3)$$, $$\varphi(x)=g(x,5)$$, and many more. The first one can be written as $$\psi(x)=g(x,3)=\cos(x^3+9)$$, while the second one is $$\varphi(x)=\cos(x^3+25)$$. They are now functions in one variable. And we can talk about $$\psi^{\prime}(x)$$ and $$\varphi^{\prime}(x)$$.

In general, if $$f$$ is a function of two variables, say $$x$$ and $$y$$, by fixing a particular value of $$y$$ we can define the function $$\psi(x)=f(x,y)$$. For this function $$\psi$$, our $$y$$ plays a role of a constant. The derivative of $$\psi$$ is a partial derivative of $$f$$. We denote it by $$f_(x,y)$$. The subscript $$x$$ emphasizes that the derivative is taken with respect to the first variable. In an analogous way we define $$f_y(x,y)$$.

Looking back at our example, $$g(x,y)=\cos(x^3+y^2)$$ we see that $$g_x(x,y)=-\sin(x^3+y^2)\cdot 3x^2$$. Similarly, $$g_y(x,y)=-\sin(x^3+y^2)\cdot 2y$$.

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