What is a Multivariate Function With Continuous Partial Derivatives

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

\(\begin{array}{l}dz = CT = \left ( \frac{\partial z}{\partial x} \right )_{at A}\Delta x + \left ( \frac{\partial z}{\partial y} \right )_{at A}\Delta y\end{array} \)

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

\(\begin{array}{l}dz = \left ( \frac{\partial z}{\partial x} \right ) dx + \left ( \frac{\partial z}{\partial y} \right )dy\end{array} \)

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

\(\begin{array}{l}df = \left ( \frac{\partial f}{\partial x_{1}} \right ) dx_{1} + \left ( \frac{\partial f}{\partial x_{2}} \right )dx_{2}+…+\left ( \frac{\partial f}{\partial x_{n}} \right ) dx_{n}\end{array} \)

where each term

\(\begin{array}{l} \frac{\partial f}{\partial x_{n}} \end{array} \)

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

\(\begin{array}{l}df=\left ( \frac{\partial f}{\partial x} \right )dx + \left ( \frac{\partial f}{\partial y} \right )dy + \left ( \frac{\partial f}{\partial z} \right )dz\end{array} \)

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

\(\begin{array}{l}\frac{du}{dt}= \frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y}\frac{dy}{dt}+…\end{array} \)

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

\(\begin{array}{l}\frac{df}{dx}= \frac{\partial f}{\partial x}\frac{dx}{dx}+\frac{\partial f}{\partial y}\frac{dy}{dx}+\frac{\partial f}{\partial z}\frac{dz}{dx}\end{array} \)

Or,

\(\begin{array}{l}\frac{df}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{dx}+\frac{\partial f}{\partial z}\frac{dz}{dx}\end{array} \)

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

\(\begin{array}{l}\frac{dw}{dx}= \frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}\frac{dy}{dx}\end{array} \)

\(\begin{array}{l}\frac{dw}{dx}= 2xy+x^{2}\frac{dy}{dx}\end{array} \)

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

\(\begin{array}{l}\frac{df}{dx}= 2x + y +x\frac{dy}{dx}+2y\frac{dy}{dx}=0\end{array} \)

\(\begin{array}{l}\Rightarrow \frac{dy}{dx}=-\frac{(2x + y)}{x+2y} \end{array} \)

Therefore,

\(\begin{array}{l}\frac{dw}{dx}= 2xy+x^{2}\left [ -\frac{(2x + y)}{x+2y} \right ]\end{array} \)

\(\begin{array}{l}\frac{dw}{dx}= 2xy-\frac{x^{2}(2x + y)}{x+2y} \end{array} \)

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

\(\begin{array}{l}\frac{dV}{dt}= \frac{\partial V}{\partial x} \frac{dx}{dt}+\frac{\partial V}{\partial y} \frac{dy}{dt}+\end{array} \)

\(\begin{array}{l}=\frac{1}{3}\pi \left [ 2xy\frac{dx}{dt}+ x^{2}\frac{dy}{dt} \right ]\end{array} \)

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

\(\begin{array}{l}\frac{dV}{dt}=\frac{1}{3}\pi [ 2.10.15(-0.3)+10^{2}(0.2)]\end{array} \)

Thus, dV/dt = – 70𝜋/3 cm³/sec

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

Frequently Asked Questions on Total Derivatives

What is the total derivative of a function?

The total derivative of multivariate function is change in the dependent variable due to a change in independent variables.

How to find the total derivative of a function?

The total derivative of a function f(x, y, z) is given by df = f_x dx + f_y dy + f_z dz, where f_x, f_y, and f_z are partial derivative of f with respect to x, y and z respectively.

What is meant by the total derivative of z if z = f(x, y)?

The total derivative of z is given by dz = f_x dx + f_y dy, where f_x and f_y are partial derivatives of f with respect to x and y respectively.

What is the difference between partial derivatives and total derivatives?

Partial derivatives are the measure of change in a function with respect to change in a single variable, while taking all other variables as constant. However, total derivative is the measure of change in the function with respect to the change in all variables.

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