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

## Practice problems

**Problem** 1. Let \(f(x,y)=xy^2+x^2y\). Find \(f_y(7,1)\).

We first find \(f_y(x,y)=2xy+x^2\), and then we put values \(7\) and \(1\) for \(x\) and \(y\) to obtain: _br__br_ \[f_y(7,1)=2\cdot 1\cdot 7+7^2=63.\]

**Problem** 2. Let \(f(x,y)=x^3+2xe^y+xy\). Find \(f_y(7,1)\).

We first find \(f_y(x,y)=2xe^y+x\), and then we put values \(7\) and \(1\) for \(x\) and \(y\) to obtain: _br__br_ \[f_y(7,1)=2\cdot 7\cdot e+7=14e+7.\]

**Problem** 3. Let \(f(x,y)=x^2+2xy^3+xy\). Find \(f_x(3,5)\).

We first find \(f_x(x,y)=2x+2y^3+y\), and then we put values \(3\) and \(5\) for \(x\) and \(y\) to obtain: _br__br_ \[f_x(3,5)=2\cdot 3+2\cdot 5^3+5=6+250+5=261.\]

**Problem** 4. Let \(f(x,y)=\frac{x+y}{x-y}\). Find \(f_x(2,3)\).

We first find \(f_x(x,y)=-\frac{2y}{(x-y)^2}\), and then we put the values \(2\) and \(3\) for \(x\) and \(y\) to obtain: _br__br_ \[f_x(2,3)=\frac{-2\cdot 3}{(2-3)^2}=-6.\]

**Problem** 5. Let \(f(x,y)=\frac{x+y}{x^2-y}\). Find \(f_x(x,y)\).

Using the quotient rule we obtain _br__br_ \[f_x(x,y)=\frac{-x^2-2xy-y}{(x^2-y)^2}.\]