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