Introduction to the Total Derivative

The total derivative of a multivariable function is the calculations of its total derivative with respect to a dependent variable. It is used when a function is defined as a function of more than one independent variable. Let’s understand how to calculate the total derivative of a function and how it differs from the partial derivative.

Understanding of the Total Derivative

The total derivative is a fundamental concept in calculus. It is used to calculate derivatives of a multivariable function that has more than one independent variable on which a dependent variable depends. Sometimes the total derivative is referred to as partial derivatives of a function. The total derivative of a function states that,

“If F=f(x,y) and x, y depend on the independent variable t, then the total derivative is the sum of the derivatives of F with respect to independent variables times derivative of all independent variables (x, y) with respect to t.”

The total derivative is used when the dependent variables are defined as the function of an independent variable.

Total Derivative Formula

The formula to calculate total derivative of a function can be generalized according to the number of dependent variables involved. For a function u = f(x, y, z) where x = x(t) and y = y(t), the total derivative formula is represented as;

$\frac{du}{dt}=\frac{\partial u}{\partial x}\times\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\times\frac{\partial y}{\partial t}$

Where, $\frac{\partial u}{\partial x}$ is the derivative of u with respect to x, $\frac{\partial u}{\partial y}$ is the derivative of u with respect to y. The sum of derivatives of u with respect to x and y is also known as the partial derivative of u = f(x, y). The above formula can also be represented as;

$\frac{du}{dt}=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$

Where,

$dx=\frac{\partial x}{\partial t}\quad\text{and}\quad dy=\frac{\partial y}{\partial t}$

Total Derivative Formula for three variables

If u is a function of three variables x, y and z, where x, y and z depends on an independent variable t, then the total derivative formula of u=f(x, y, z) is represented as;

$\frac{du}{dt}=\frac{\partial u}{\partial x}\times\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\times \frac{\partial y}{\partial t}+\frac{\partial u}{\partial z}\times\frac{\partial z}{\partial t}$

Where, $\frac{\partial u}{\partial x}$ is the derivative of u with respect to x, $\frac{\partial u}{\partial y}$ is the derivative of u with respect to y and $\frac{\partial u}{\partial z}$ is the derivative of u with respect to z. The above formula can also be represented as,

$\frac{du}{dt}=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz$

How to find the total derivative of a function?

The total derivative of a function can be calculated by using the total derivative formula. For this, we have to calculate all partial derivatives of the function according to the variables involved. The following steps can be used to find the total derivative of a function.

1. Identify the dependent and independent variables of the given function for which we want to calculate total derivative. For example, we have to find the derivative of x^3y^3z^3. 
2. It is clear that the function u=f(x, y, z) = x^3y^3z^3 has three independent variables x, y, z and one dependent function u. 
3. Now calculate all of the partial derivatives involved in the total derivative formula. 
4. Substitute the derivatives calculated in step 3 in the total derivative formula. 
5. Simplify if needed.

Let's understand how to calculate the total derivative of a function in the following example. 

Total Derivative Example 

Suppose a function u = sin x/y where x = et and y = t2. Calculate the derivative of u.

Since the given function is a multivariable function, we need to calculate the derivative of u with respect to t. For this, we will use the total derivative formula. 

$\frac{du}{dt}=\frac{\partial u}{\partial x}\times\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\times\frac{\partial y}{\partial t}$

Let’s calculate all partial derivatives one-by-one first.

$\frac{\partial u}{\partial x}=\frac{\partial}{\partial x}(\sin\frac{x}{y})$

Here, we are calculating derivatives with respect to x, so we will treat y as a constant.

$\frac{\partial u}{\partial x}=\cos\frac{x}{y}\times\frac{1}{y}$

$\frac{\partial u}{\partial y}=\frac{\partial}{\partial y}(\sin\frac{x}{y})=\cos\frac{x}{y}\times\frac{\partial}{\partial y}(\frac{x}{y})$

We get, 

$\frac{\partial u}{\partial y}=\frac{\partial}{\partial y}(\sin\frac{x}{y})=-\frac{x}{y^2}\cos\frac{x}{y}$

Similarly, 

$\frac{\partial x}{\partial t}=\frac{\partial}{\partial t}(e^t)=e^t$

$\frac{\partial y}{\partial t}=\frac{\partial}{\partial t}(t^2)=2t$

Substituting the values of all derivatives in the total derivative formula,

$\frac{du}{dt}=\frac{\partial u}{\partial x}\times\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\times\frac{\partial y}{\partial t}$

$\frac{du}{dt}=\frac{1}{y}\cos\frac{x}{y}\times e^t-\frac{x}{y^2}\cos\frac{x}{y}\times2t$

Simplifying, 

$\frac{du}{dt}=\frac{1}{y}\cos\frac{x}{y}\left(e^t-\frac{x}{y}2t\right)$

Which is the derivative of the given function.

Comparison between total derivative and partial derivative 

The comparison between the total derivative and partial derivative can be easily analysed using the following difference table.

Conclusion

In calculus, a concept which is used to calculate derivatives of a function containing more than one independent variable, is called total derivative. It is based on the concept of partial derivatives. It has many applications in mathematics, because we can calculate the rate of change in a quantity or a function with respect to different factors by using total derivatives. Hence it has a great importance in differential calculus.