PARTIAL DERIVATIVES: NOTATION, EXAMPLES AND SOLVED EXERCISES - MATHS - 2025

The partial derivatives of a function of several variables are those that determine the rate of change of the function when one of the variables has an infinitesimal variation, while the other variables remain unchanged.

To make the idea more concrete, suppose the case of a function of two variables: z = f (x, y). The partial derivative of the function f with respect to the variable x is calculated as the ordinary derivative with respect to x, but taking the variable y as if it were constant.

Figure 1. Function f (x, y) and its partial derivatives ∂ _x f y ∂ _y f at point P. (Elaborated by R. Pérez with geogebra)

Partial derivative notation

The partial derivative operation of the function f (x, y) on the variable x is denoted in any of the following ways:

In partial derivatives the symbol ∂ (a kind of rounded letter d also called Jacobi's d) is used, as opposed to the ordinary derivative for single-variable functions where the letter d is used for derivative.

In general terms, the partial derivative of a multivariate function, with respect to one of its variables, results in a new function in the same variables of the original function:

∂ _x f (x, y) = g (x, y)

∂ _y f (x, y) = h (x, y).

Calculation and meaning of the partial derivative

To determine the rate of change or slope of the function for a specific point (x = a, y = b) in the direction parallel to the X axis:

1- The function ∂ _x f (x, y) = g (x, y) is calculated, taking the ordinary derivative in the variable x and leaving the variable y fixed or constant.

2- Then the value of the point x = a and y = b is substituted in which we want to know the rate of change of the function in the x direction:

{Slope in the x direction at the point (a, b)} = ∂ _x f (a, b).

3- To calculate the rate of change in the y direction at the coordinate point (a, b), first calculate ∂ _and f (x, y) = h (x, y).

4- Then the point (x = a, y = b) is substituted in the previous result to obtain:

{Slope in the y direction at the point (a, b)} = ∂ _y f (a, b)

Examples of partial derivatives

Some examples of partial derivatives are as follows:

Example 1

Given the function:

f (x, y) = -x ^ 2 - y ^ 2 + 6

Find the partial derivatives of the function f with respect to the variable x and the variable y.

Solution:

∂ xf = -2x

∂ yf = -2y

Note that to calculate the partial derivative of the function f with respect to the variable x, the ordinary derivative with respect to x was carried out, but the variable y was taken as if it were constant. Similarly, in the calculation of the partial derivative of f with respect to y, the variable x has been taken as if it were a constant.

The function f (x, y) is a surface called a paraboloid shown in figure 1 in ocher color.

Example 2

Find the rate of change (or slope) of the function f (x, y) from Example 1, in the direction of the X-axis and the Y-axis for the point (x = 1, y = 2).

Solution: To find the slopes in the x and y directions at the given point, simply substitute the values ​​of the point into the function ∂ _x f (x, y) and into the function ∂ _y f (x, y):

∂ _x f (1,2) = -2⋅ 1 = -2

∂ _and f (1,2) = -2⋅ 2 = -4

Figure 1 shows the tangent line (in red color) to the curve determined by the intersection of the function f (x, y) with the plane y = 2, the slope of this line is -2. Figure 1 also shows the tangent line (in green) to the curve that defines the intersection of the function f with the plane x = 1; This line has slope -4.

Exercises

Exercise 1

A conical glass at a given time contains water so that the surface of the water has radius r and depth h. But the glass has a small hole in the bottom through which water is lost at a rate of C cubic centimeters per second. Determine the rate of descent from the water surface in centimeters per second.

Solution:

First of all, it is necessary to remember that the volume of water at the given instant is:

Volume is a function of two variables, radius r and depth h: V (r, h).

When the volume changes by an infinitesimal amount dV, the radius r of the water surface and the depth h of the water also change according to the following relationship:

dV = ∂ _r V dr + ∂ _h V dh

We proceed to calculate the partial derivatives of V with respect to r and h respectively:

∂ _r V = ∂ _r (⅓ π r ^ 2 h) = ⅔ π rh

∂ _h V = ∂ _h (⅓ π r ^ 2 h) = ⅓ π r ^ 2

Furthermore, the radius r and the depth h meet the following relationship:

Dividing both members by the time differential dt gives:

dV / dt = π r ^ 2 (dh / dt)

But dV / dt is the volume of water lost per unit of time that is known to be C centimeters per second, while dh / dt is the rate of descent of the free surface of water, which will be called v. That is, the water surface at the given instant descends at a speed v (in cm / s) given by:

v = C / (π r ^ 2).

As a numerical application, suppose that r = 3 cm, h = 4 cm, and the leak rate C is 3 cm ^ 3 / s. Then the speed of descent of the surface at that instant is:

v = 3 / (π 3 ^ 2) = 0.11 cm / s = 1.1 mm / s.

Exercise 2

The Clairaut-Schwarz theorem states that if a function is continuous in its independent variables and its partial derivatives with respect to the independent variables are also continuous, then the second-order mixed derivatives can be interchanged. Check this theorem for the function

f (x, y) = x ^ 2 y, that is, it must be true that f _xy f = ∂ _yx f.

Solution:

∂ _xy f = ∂ _x (∂ _y f) while ∂ _yx f = ∂ _y (∂ _x f)

∂ _x f = 2 xy; ∂ _y f = x ^ 2

∂ _xy f = ∂ _x (∂ _y f) = 2x

∂ _yx f = ∂ _y (∂ _x f) = 2x

Schwarz's theorem has been proven to hold, since the function f and its partial derivatives are continuous for all real numbers.

References

1. Frank Ayres, J., & Mendelson, E. (2000). Calculation 5ed. Mc Graw Hill.
2. Leithold, L. (1992). The calculation with analytic geometry. HARLA, SA
3. Purcell, EJ, Varberg, D., & Rigdon, SE (2007). Calculation. Mexico: Pearson Education.
4. Saenz, J. (2005). Diferential calculus. Hypotenuse.
5. Saenz, J. (2006). Integral calculus. Hypotenuse.
6. Wikipedia. Partial derivative. Recovered from: es.wikipedia.com