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, \( g_y(x,y)=-\sin(x^3+y^2)\cdot 2y \).


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us