# 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

**partial derivative**of f with respect to y which is denoted by

We might also use the limits to define partial derivatives of function f as follows:

## 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

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

Now __assume x is constant__and differentiate with respect to y to obtain

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

### Example 2

Find 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

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)

Differentiate with respect to y assuming x is constant

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)

### 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.

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}

Differentiate with respect to y assuming x is constant.

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}

### 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

f_x = \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\ln(x^2+2y)) \\\\
= \frac{\partial}{\partial x}(x^2+2y) \cdot \dfrac{1}{x^2+2y} = \dfrac{2x}{x^2+2y}

Differentiate with respect to y

f_y = \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\ln(x^2+2y)) \\\\
= \frac{\partial}{\partial y}(x^2+2y) \cdot \dfrac{1}{x^2+2y} = \dfrac{2}{x^2+2y}

### 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

1. f_{x}=(x + 1)e

^{x + y}, f

_{y}= x e

^{x + y}

2. f

_{x}= 1 / x , f

_{y}= 1 / (y + 2)

3. f

_{x}= x cos (x - y) + sin (x - y) , f

_{y}= -x cos (x - y)

## More References and Links to Partial Derivatives and Mtlivariable Functions

Partial Derivative CalculatorTables of Formulas for Derivatives

Rules of Differentiation of Functions in Calculus

Multivariable Functions

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