Partial Derivative

Partial Derivative 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:

Example# 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
fx(x,y) = 2x y
f_y(x,y) = x² + 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) = 2² + 2 = 6

Phrasing and notation

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

• “The partial derivative of ‘with respect to”

• “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:

fx ↔∂x/∂f

fy ↔ ∂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 (x0) = h→0lim ( hf (x0 +h)−f(x0 )) / h

• h represents the “tiny value” that we intuitively think of as dx.
• The h→0 under the limit indicates that we care about very small values of h, those approaching 0.
• f(x0+h)−f(x0) is the change in the output that results from addinghto the input, which is what we think of asdf.

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

∂x / ∂f (x0 ,y0 ,…) = h→0lim (hf (x0 +h, y0 ,…) − f (x0 ,y0 ,…)) /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.

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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 = ∂²f / ∂x² = ∂(∂f / ∂x) / ∂x
= ∂(∂[ sin (x y) ]/ ∂x) / ∂x
= ∂(y cos (x y) ) / ∂x
= – y² sin (x y) )
f_yy can be calculated as follows:
f_yy = ∂²f / ∂y² = ∂(∂f / ∂y) / ∂y
= ∂(∂[ sin (x y) ]/ ∂y) / ∂y
= ∂(x cos (x y) ) / ∂y
= – x² sin (x y)

Example# 2
Find f_xx, f_yy, f_xy, f_yx given that f(x , y) = x³ + 2 x y.
Solution:
f_xx is calculated as follows:
f_xx = ∂²f / ∂x² = ∂(∂f / ∂x) / ∂x
= ∂(∂[ x³ + 2 x y ]/ ∂x) / ∂x
= ∂( 3 x² + 2 y ) / ∂x
= 6x
f_yy is calculated as follows:
f_yy = ∂²f / ∂y² = ∂(∂f / ∂y) / ∂y
= ∂(∂[ x³ + 2 x y ]/ ∂y) / ∂y
= ∂( 2x ) / ∂y
= 0
f_xy is calculated as follows:
f_xy = ∂²f / ∂y∂x = ∂(∂f / ∂x) / ∂y
= ∂(∂[ x³ + 2 x y ]/ ∂x) / ∂y
= ∂( 3 x² + 2 y ) / ∂y
= 2
f_yx is calculated as follows:
f_yx = ∂²f / ∂x∂y = ∂(∂f / ∂y) / ∂x
= ∂(∂[ x³ + 2 x y ]/ ∂y) / ∂x
= ∂( 2x ) / ∂x
= 2

Example# 3
Find f_xx, f_yy, f_xy, f_yx given that f(x , y) = x³y⁴ + x² y.
Solution:
f_xx is calculated as follows:
f_xx = ∂²f / ∂x² = ∂(∂f / ∂x) / ∂x
= ∂(∂[ x³y⁴ + x² y ]/ ∂x) / ∂x
= ∂( 3 x²y⁴ + 2 x y) / ∂x
= 6x y⁴ + 2y
f_yy is calculated as follows:
f_yy = ∂²f / ∂y² = ∂(∂f / ∂y) / ∂y
= ∂(∂[ x³y⁴ + x² y ]/ ∂y) / ∂y
= ∂( 4 x³y³ + x² ) / ∂y
= 12 x³y^2
f_xy is calculated as follows:
f_xy = ∂²f / ∂y∂x = ∂(∂f / ∂x) / ∂y
= ∂(∂[ x³y⁴ + x² y ]/ ∂x) / ∂y
= ∂( 3 x²y⁴ + 2 x y ) / ∂y
= 12 x²y³ + 2 x
f_yx is calculated as follows:
f_yx = ∂²f / ∂x∂y = ∂(∂f / ∂y) / ∂x
= ∂(∂[ x³y⁴ + x² y ]/ ∂y) / ∂x
= ∂(4 x³y³ + x²) / ∂x
= 12 x²y³ + 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).