Total Derivative

The total derivative of a function of several variables means the total change in the dependent variable due to the changes in all the independent variables. Suppose z = f(x, y) be a function of two variables, where z is the dependent variable and x and y are the independent variables. The total derivative of f with respect to x and y will be the total change in z due to a change in both x and y. However, the change in z is only a linear approximation of the actual change in z.

We shall understand the total derivative of a function by its geometrical interpretation. Also, understand the differentiability of a function by its total derivative.

Geometrical Interpretation of Total Derivatives

Consider a single variable function y = f(x); the total derivative of the function is given by

dy = f’(x) Δx

this quantity determines the approximate change in f(x) due to the change in x from x to x + Δx, as shown in the figure below

In the figure, Δy = CB = (y + Δy) – y = f(x + Δx) – f(x) as Δy → 0 ⇒ Δy = dy

And dy = AP × CA = f’(x) Δx

where Δx is very small change in x.

In the case of a function of two variables, z = f(x, y), let A(x, y, z) be any point on the surface of f as shown in the figure below. If Δx and Δy are small changes in x and y respectively. The change produced in z is given by Δz = CB = f(x + Δx, y + Δy) – f(x, y).

An approximation to Δz, that is, dz is given by

where Δx and Δy are small changes in x and y. T is a point in the tangent plane at A, as shown in figure 2.

As Δx → 0 and Δy → 0, Δx ≈ dx and Δy ≈ dy

Thus, total derivative of function z = f(x, y) is given as

Similarly, the total derivative of function of three or more variables (x₁, x₂, …, x_n) is given by

where each term

is the partial derivative of f with respect to x_i for i = 1, 2, 3, …, n.

Definition of Total derivative of a Function

The total derivative of a function f(x, y, z) is given by

where x, y and z may or may not be independent of each other.

Chain Rule for Total Derivatives

Let u = f(x, y, …) be a continuous function of several variables x, y, … with continuous partial derivatives u_x, u_y, … If each variable is a function t, that is, x = x(t), y = y(t), and so on.

Then the total derivative of u with respect to t is given by

This is known as Chain Rule for total derivatives.

If u = f(x, y, z) be such that y and z are function of x. Then f is a function of one independent variable. Then the total derivative of f is given by

Or,

Related Articles

• Partial Derivatives
• Chain Rule of Differentiation
• Limits
• Continuity
• Independent and Dependent Variables

Solved Examples on Total Derivatives

Example 1:

Find the total differential coefficient of the function x²y with respect to x where x^2.+ xy + y² = 1.

Solution:

Let w = x²y, we have to find the total differential coefficient of w with respect to x, that is, dw/dx

We have the total derivative of w as

dw = w_x dx + w_y dy where w_x and w_y are partial derivatives of w with respect to x and y respectively.

Then the total differential coefficient of w is

We have to find the value of dy/dx.

Let f = x^2.+ xy + y² = 1

Differentiating both sides with respect to x, we get

Therefore,

Example 2:

The altitude of a right circular cone is 15 cm and is increasing at the rate 0.2 cm/sec. The radius of the base is 10 cm and is decreasing at the rate 0.3 cm/sec. How fast the volume of the cone is changing.

Solution:

Let x be the base radius and y be the altitude of the cone.

Then volume of the cone V = ⅓ 𝜋 x²y

Then rate of change in volume is given by

Now, given x = 10 cm, y = 15 cm, dx/dt = – 0.3 cm/sec and dy/dt = 0.2 cm/sec

Thus, dV/dt = – 70𝜋/3 cm³/sec

That is, volume decreasing at the rate of 70𝜋/3 cm³/sec.