**Definition:**

**Partial derivatives** are **defined** as **derivatives** of a function of multiple variables when all but the variable of interest is held fixed during the **differentiation**.

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

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

Examples:

**E****xample# 1:**

Find the partial derivatives f_{x} and f_{y} if f(x , y) is given by

**Solution of example# 1:**

Assume ‘y’ is constant and differentiate with respect to x to obtain

Now assume x is constant and differentiate with respect to y to obtain

**Example# 2:**

Find f_{x} and f_{y} if f(x , y) is given by

**Solution to Example 2:**

Differentiate with respect to x assuming y is constant

Differentiate with respect to y assuming x is constant

**Example# 3:**

Find f_{x} and f_{y} if f(x , y) is given by

**Solution to Example 3:**

Differentiate with respect to x assuming y is constant using the product rule of differentiation.

Differentiate with respect to y assuming x is constant.

**Example# 4:**

Find f_{x} and f_{y} if f(x , y) is given by

**Solution to Example 4:**

Differentiate with respect to x to obtain

Differentiate with respect to y

**Example# 5:**

Find f_{x}(2 , 3) and f_{y}(2 , 3) if f(x , y) is given by

**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

**Phrasing and notation**

Here are some of the phrases you might hear in reference to **∂***f **/***∂***f*** **operation:

- “The partial derivative of
**‘***￼*”*￼*

- “Del
**f,**del**x**” - “Partial
**f**, partial**x**” - “The partial derivative (of ‘
*￼*) in the ‘*￼*-direction”

*Alternate notation:*

In the same way that people sometimes prefer to write *f ***′** instead of **d ***f** **/** ***d*** x*** , **we have the following notation:

* **f**x* ↔∂*x/*∂*f*

* **f**y* ↔ ∂*f/ *∂*y*

* **F**(**some variable) *↔ ∂*f**/ *∂ (*That same variable*)

A more formal definition:

Although thinking of *dx* or ∂*x* as really tiny changes in the value of *x* is a useful intuition, it is healthy to occasionally step back and remember that defining things precisely requires introducing limits. After all, what specific small value would ∂*x* be? One one hundredth? One one millionth? 10^10^10?

The point of calculus is that we don’t use any one tiny number, but instead consider all possible values and analyze what tends to happen as they approach a limiting value. The single variable derivative, for example, is defined like this:

*dx**/**df* (*x**0) =* *h*→0lim ( *h**f *(*x*0 +*h*)−*f*(*x*0 )) **/ ***h*

represents the “tiny value” that we intuitively think of as*h*.*dx*- The
*h***→0**under the limit indicates that we care about very small values of, those approaching 0.*h* *￼**f***(***x***0****+***h***)−***f***(***x***0****)**is the change in the output that results from adding*h***d***f**.*

Formally defining the partial derivative looks almost identical. If *￼**x*,*y*…), is a function with multiple inputs, here’s how that looks:

∂*x *** / **∂

*f*(

*x*0 ,

*y*0 ,…) =

*h*→0lim (

*h*

*f*(

*x*0 +

*h*,

*y*0 ,…) −

*f*(

*x*0 ,

*y*0 ,…)) /

*h*

The point is that *h*, which represents a tiny tweak to the input, is added to different input variables depending on which partial derivative we are taking.

People will often refer to this as the **limit definition** of a partial derivative.

**Second Partial Derivative:**** **

A brief overview of second partial derivative, the symmetry of mixed partial derivatives, and higher order partial derivatives.

**Notations of Second Order Partial Derivatives:**

For a two variable function f(x , y), we can define 4 second order partial derivatives along with their notations.

**Examples with Detailed solutions:**

**Example# 1**

Find f_{xx}, f_{yy} given that f(x , y) = sin (x y) **Solution**:

f_{xx} may be calculated as follows:

f_{xx} = ∂^{2}f / ∂x^{2} = ∂(∂f / ∂x) / ∂x

= ∂(∂[ sin (x y) ]/ ∂x) / ∂x

= ∂(y cos (x y) ) / ∂x

= – y^{2} sin (x y) )

f_{yy} can be calculated as follows:

f_{yy} = ∂^{2}f / ∂y^{2} = ∂(∂f / ∂y) / ∂y

= ∂(∂[ sin (x y) ]/ ∂y) / ∂y

= ∂(x cos (x y) ) / ∂y

= – x^{2} sin (x y)

**Example# 2**

Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3} + 2 x y. **Solution**:

f_{xx} is calculated as follows:

f_{xx} = ∂^{2}f / ∂x^{2} = ∂(∂f / ∂x) / ∂x

= ∂(∂[ x^{3} + 2 x y ]/ ∂x) / ∂x

= ∂( 3 x^{2} + 2 y ) / ∂x

= 6x

f_{yy} is calculated as follows:

f_{yy} = ∂^{2}f / ∂y^{2} = ∂(∂f / ∂y) / ∂y

= ∂(∂[ x^{3} + 2 x y ]/ ∂y) / ∂y

= ∂( 2x ) / ∂y

= 0

f_{xy} is calculated as follows:

f_{xy} = ∂^{2}f / ∂y∂x = ∂(∂f / ∂x) / ∂y

= ∂(∂[ x^{3} + 2 x y ]/ ∂x) / ∂y

= ∂( 3 x^{2} + 2 y ) / ∂y

= 2

f_{yx} is calculated as follows:

f_{yx} = ∂^{2}f / ∂x∂y = ∂(∂f / ∂y) / ∂x

= ∂(∂[ x^{3} + 2 x y ]/ ∂y) / ∂x

= ∂( 2x ) / ∂x

= 2

**Example# 3**

Find f_{xx}, f_{yy}, f_{xy}, f_{yx} given that f(x , y) = x^{3}y^{4} + x^{2} y. **Solution****:**

f_{xx} is calculated as follows:

f_{xx} = ∂^{2}f / ∂x^{2} = ∂(∂f / ∂x) / ∂x

= ∂(∂[ x^{3}y^{4} + x^{2} y ]/ ∂x) / ∂x

= ∂( 3 x^{2}y^{4} + 2 x y) / ∂x

= 6x y^{4} + 2y

f_{yy} is calculated as follows:

f_{yy} = ∂^{2}f / ∂y^{2} = ∂(∂f / ∂y) / ∂y

= ∂(∂[ x^{3}y^{4} + x^{2} y ]/ ∂y) / ∂y

= ∂( 4 x^{3}y^{3} + x^{2} ) / ∂y

= 12 x^{3}y^{2} ^{f}_{xy} is calculated as follows:

f_{xy} = ∂^{2}f / ∂y∂x = ∂(∂f / ∂x) / ∂y

= ∂(∂[ x^{3}y^{4} + x^{2} y ]/ ∂x) / ∂y

= ∂( 3 x^{2}y^{4} + 2 x y ) / ∂y

= 12 x^{2}y^{3} + 2 x

f_{yx} is calculated as follows:

f_{yx} = ∂^{2}f / ∂x∂y = ∂(∂f / ∂y) / ∂x

= ∂(∂[ x^{3}y^{4} + x^{2} y ]/ ∂y) / ∂x

= ∂(4 x^{3}y^{3} + x^{2}) / ∂x

= 12 x^{2}y^{3} + 2x

The gradient

The gradient stores all the partial derivative information of a multivariable function. But it’s more than a mere storage device, it has several wonderful interpretations and many, many uses.

Example:

The **gradient** of a function *f*, denoted as ∇*f,* is the collection of all its partial derivatives into a vector. This is most easily understood with an example.

Let f(x,y)= x^2y. Find ∇f(3,2).

**Solution:** The gradient is just the vector of partial derivatives. The partial derivatives of f, at the point (x,y)=(3,2) are:

∂f/∂x (x,y) = 2xy

∂f/∂x (3,2) = 12

∂f/∂y (x,y) = x2

∂f/∂y (3,2) = 9

Therefore, the gradient is

∇f(3,2) = 12i+9j = (12,9).