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Partial Derivatives

Definition of Partial Derivatives

Let

$f(x,y)$ be a function with two variables. If we keep

$y$ constant and differentiate

$f$ (assuming

$f$ is differentiable) with respect to the variable

$x$ , using the rules and formulas of differentiation, we obtain what is called the partial derivative of

$f$ with respect to

$x$ which is denoted by

$\frac{\partial f}{\partial x} \; \text{or} \; f_x$ Similarly If we keep

$x$ constant and differentiate

$f$ (assuming

$f$ is differentiable) with respect to the variable

$y$ , we obtain what is called the partial derivative of

$f$ with respect to

$y$ which is denoted by

$\frac{\partial f}{\partial y} \; \text{or} \; f_y$ We might also use the limits to define partial derivatives of function

$f$ as follows:

$\frac{\partial f}{\partial x} = \lim_{h\to 0} \frac{f(x+h,y)-f(x,y)}{h}$ and

$\frac{\partial f}{\partial y} = \lim_{k\to 0} \frac{f(x,y+k)-f(x,y)}{k}$

Examples with Detailed Solutions

We now present several examples with detailed solution on how to calculate partial derivatives.

Example 1

Find the partial derivatives

$f_x$ and

$f_y$ if

$f(x , y)$ is given by

$f(x,y) = x^2 y + 2 x + y$ Solution to Example 1:
Assume

$y$ is constant and differentiate with respect to

$x$ to obtain

$\begin{align*} f_x &= \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 y + 2 x + y ) \\ &= \frac{\partial}{\partial x}(x^2 y ) + \frac{\partial}{\partial x}(2 x) + \frac{\partial}{\partial x}( y ) = 2 xy + 2 + 0 = 2xy + 2 \end{align*}$ Assume

$x$ is constant and differentiate with respect to

$y$ to obtain

$\begin{align*} f_y &= \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2 y + 2 x + y ) \\ &= \frac{\partial}{\partial y}(x^2 y ) + \frac{\partial}{\partial y}(2 x) + \frac{\partial}{\partial y}( y ) = x^2 + 0 + 1 = x^2 + 1 \end{align*}$

Example 2

Find the partial derivatives

$f_x$ and

$f_y$ if

$f(x , y)$ is given by

$f(x,y) = \sin(x y) + \cos x$ Solution to Example 2:
Differentiate with respect to

$x$ assuming

$y$ is constant

$\begin{align*} f_x &= \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\sin(x y) + \cos x ) \\ &= \frac{\partial}{\partial x}(\sin(x y) ) + \frac{\partial}{\partial x}(\cos x) = y \cos(x y) -\sin(x) \end{align*}$ Differentiate with respect to

$y$ assuming

$x$ is constant

$\begin{align*} f_y &= \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\sin(x y) + \cos x ) \\ &= \frac{\partial}{\partial y}(\sin(x y) ) + \frac{\partial}{\partial y}(\cos x) = x \cos(x y) - 0 = x \cos(x y) \end{align*}$

Example 3

Find

$f_x$ and

$f_y$ if

$f(x , y)$ is given by

$f(x,y) = x e^{x y}$ Solution to Example 3:
Differentiate with respect to

$x$ assuming

$y$ is constant using the product rule of differentiation.

$\begin{align*} f_x &= \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x e^{x y}) \\ &= \frac{\partial}{\partial x}(x) e^{x y} + x \frac{\partial}{\partial x}(e^{x y}) = 1 \cdot e^{x y} + x \cdot y e^{x y} = (1+xy) e^{x y} \end{align*}$ Differentiate with respect to

$y$ assuming

$x$ is constant.

$\begin{align*} f_y &= \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x e^{x y}) \\ &= x \frac{\partial}{\partial y}(e^{x y}) = x \cdot x e^{x y} = x^2 e^{x y} \end{align*}$

Example 4

Find

$f_x$ and

$f_y$ if

$f(x , y)$ is given by

$f(x,y) = \ln(x^2+2y)$ Solution to Example 4:
Differentiate with respect to

$x$ to obtain

$\begin{align*} f_x &= \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\ln(x^2+2y)) \\ &= \frac{\partial}{\partial x}(x^2+2y) \cdot \frac{1}{x^2+2y} = \frac{2x}{x^2+2y} \end{align*}$ Differentiate with respect to

$y$

$\begin{align*} f_y &= \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\ln(x^2+2y)) \\ &= \frac{\partial}{\partial y}(x^2+2y) \cdot \frac{1}{x^2+2y} = \frac{2}{x^2+2y} \end{align*}$

Example 5

Find

$f_x(2 , 3)$ and

$f_y(2 , 3)$ if

$f(x , y)$ is given by

$f(x,y) = y x^2 + 2 y$ Solution to Example 5:
We first find the partial derivatives

$f_x$ and

$f_y$

$f_x(x,y) = 2x y$

$f_y(x,y) = x^2 + 2$ We now calculate

$f_x(2 , 3)$ and

$f_y(2 , 3)$ by substituting

$x$ and

$y$ by their given values

$f_x(2,3) = 2 (2)(3) = 12$

$f_y(2,3) = 2^2 + 2 = 6$

Exercises

Find partial derivatives

$f_x$ and

$f_y$ of the following functions
1.

$f(x , y) = x e^{x + y}$
2.

$f(x , y) = \ln ( 2 x + y x)$
3.

$f(x , y) = x \sin(x - y)$

Answers to the Above Exercises

$f_x =(x + 1)e^{x + y} \quad$ ,

$\quad f_y = x e^{x + y}$
2.

$f_x = 1 / x \quad$ ,

$\quad f_y = 1 / (y + 2)$
3.

$f_x = x \cos (x - y) + \sin (x - y) \quad$ ,

$\quad f_y = -x \cos (x - y)$

Partial Derivatives

Definition of Partial Derivatives

Examples with Detailed Solutions

Example 1

Example 2

Example 3

Example 4

Example 5

Exercises

Answers to the Above Exercises

More References and Links to Partial Derivatives and Mtlivariable Functions