Partial Derivative Calculator

Partial Derivative Calculator

Find the first partial derivative of multivariable functions using the partial derivative calculator. This multivariable derivative calculator will give results with steps. Moreover, this partial derivative calculator accepts functions with 3 variables as well. 

What are partial derivatives?

When the differentiation is carried out on a function having more than one variable, the result is a partial derivative. The process is called partial differentiation. It is similar to actual derivatives just the object (function) changes. 

Where in simple differentiation, derivatives are found with respect to the variable of the function, here, due to the presence of more than one variable, the variable has to be chosen for differentiation. 

Partial derivatives are required when the change with respect to one variable has to be noticed. The second variable is considered constant. In a multivariable function, each variable is independent. 

 
For example, there is a point (1,2) for a function x² +3y². If this function was to be differentiated with respect to x, it would mean keeping the y constant and moving the x as little as delta (a tiny change) may be to (1.001, 2).

How to find the partial derivatives?

In this type of differentiation, the same set of rules is used e.g. the power rule, quotient rule, etc the only difference comes when you have to deal with the other variable.  

The above partial derivative calculator is the most suitable option for this purpose as differentiation can be tricky. But the manual points to stick by are:

1. Apply the partial derivative notation to the function i.e ∂f/∂x
2. Use the variable with which the differentiation is required, in the derivative notation. 
3. Use the rules of differentiation. 
4. Treat the other variables as constant (as they are not changing).

When a different variable (in the upcoming example y) comes separately like ∂f/∂x 2y², the answer is zero. because honestly there is no change in y with respect to x. Both are independent. 

But when it comes to the main variable such as ∂f/∂x 2y2x2, the variable is kept unchanged. If it is solved to zero, the whole value will eventually become zero. This would suggest no change in x as well which is not true. 

Example:

Find the partial derivative of the multivariable function 10x⁴ − 18xy² + 10y³ with respect to y.

Solution:

Step 1: Apply the derivative notation.

= ∂f/∂y (10x⁴) - ∂f/∂y(18xy²) + ∂f/∂y (10y³)

Step 2: Apply the constant rule on the first value.

=  ∂f/∂y (10x⁴) - ∂f/∂y(18xy²) + ∂f/∂y (10y³)

=  0 - ∂f/∂y(18xy2) + ∂f/∂y (10y3)

Step 3: Apply the power rule to the next two values. 

= - (18.2xy^2-1) + (10.3y^3-1)

=  - 36xy + 30y²

= 6y(5y - 6x)