Partial derivatives value using a contour map

 Goal: Estimate the value of partial derivatives using 

Partial derivative value using a contour map

We can estimate the value of partial derivatives which represent the slope of a tangent line by using the slope of a secant line.

To estimate fₓ (x,y), use the slope of the secant line in the x direction:

ẟf/ẟx ≈Δf/Δx

To estimate f_y(x,y) use the slope of the secant line in the y direction:

ẟf/ẟy ≈Δf/Δy.

Example:

Solution
The graph of the function g is represented below:

The circle in red is given by the equation 1 = ⎷9-x²-y². When simplified, this equation becomes 
x²+y² = 8. It intercepts the x-axis at the point (2⎷2, 0). The circle in blue or the inner circle is given by the equation 2 = ⎷9-x²-y² It simplifies to x²+y² = 5. It intercepts the x-axis at the point (⎷5, 0). To estimate the partial derivative, we use the slope formula:

Let's calculate the exact value of ẟf/ẟx by calculating the partial derivative of g. First, let's rewrite

the function g as:

 

Now let's differentiate g with respect to x while holding y constant. This is done by using the chain rule

 

     

The estimate of the partial derivative corresponds to the slope of the secant line passing through the points: (2⎷2, 0, g(2⎷2, 0)) and (⎷5, 0. g(⎷5, 0) ). It represents an approximation of the slope of the tangent line to the surface at the point (⎷5, 0. g(⎷5, 0)), which is parallel to the x-axis.

Practice

Calculating partial derivatives of a function of two variables without using limit

 Goal: Calculate partial derivatives without using limit

The idea behind calculating partial derivatives is to keep all independent variables, other than the ones with respect to which you are differentiating, as constant. Then proceed as if you were using a function of single variable. To prove this let's fix y and pose g(x) = f(x, y)

The same is true when differentiating with respect to y. In this case, we fix x and pose  h(y) = f(x,y) 

as a function of y.                                                                                                                                        

                                                                                                                                               

Example

Calculate 

for the following functions by holding the opposite variable constant and differentiating

Solution

The derivative of the third, fifth and sixth terms are all zero because they do not contain the variable x. Therefore they are treated as constants. The derivative of the second term is -3y since y is considered as a constant number.

Then differentiate g(x,y) with respect to x using 

    
the chain rule and the power rule.

Then differentiate g(x,y) with respect to y using

 the chain rule and the power rule.

   

Practice

by holding the opposite variable 

constant and then differentiating

Derivatives of a function of two variables

 Goal: Calculate the partial derivatives of a function of two variables

 Derivatives of a function of two variables

We have seen that some methods of the function of one variable can be applied to the function of two variables. In the notation of Leibniz dy/dx, x is the independent variable and y is the dependent variable. In a function of two variables, how do we adapt this notation? What would be an acceptable definition for the derivative of a function of two variables. This leads to the notion of partial derivatives

Definition

Let f be a function of two variables. Then the partial derivative of with respect to x, written as    

 is defined as:  

The partial derivative of f with respect to y, written as is defined as

 

N.B The notation   is called partial derivative of f with respect to x. 

The notation  is called partial derivative of with respect to y.

Example 

Use the definition of the partial derivative as a limit to calculate  and

for the function :

 

Solution

First, let's calculate f(x+h, y):

  

Let's substitute this in the definition of the partial derivative of f with respect to x.

 

 

To calculate the partial derivative of f with respect to y, let's calculate f(x, y+h)

 

Let's substitute this in the definition of the partial derivative with respect to y:

 

Practice

Use the definition of the partial derivative as a limit to calculate and for the following function: