Caluclus: Table of contents

Partial Derivatives


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.


Consider the following function of two variables:


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 \(f_y(7,1)\).

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

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

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

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