Introduction to Partial derivative

There are different rules and methods of differentiation in calculus. One of these methods is the partial derivative. It is a method to differentiate a multivariable function, i.e., a function that depends on more than one variable. Let’s understand how to apply the partial derivative and its relationship with other derivative rules.

Understanding the Partial Derivative

The partial derivative is a derivative used to calculate the rate of change of a function depending on more than one variable. It computes the derivative of a function with respect to one variable and treats the other variable as a constant. By definition, the partial derivative is defined as;

“The derivative of function f(x,y) is equal to the sum of derivative f(x.y) with respect to x and with respect to y, is known as the partial derivative of f(x,y).”

In other words, finding the derivative of y=f(x,y) with respect to x and y one by one and then adding them together gives the partial derivative of y=f(x,y).

Partial Derivative Formula

The partial derivative tells how a function changes with the change in both independent variables. For a function y = f(x,y), the partial derivative formula is expressed as:

$f'(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$

Where,

• $\frac{\partial f}{\partial x}$ is the derivative of f(x,y) with respect to x.
• $\frac{\partial f}{\partial y}$ is the derivative of f(x,y) with respect to y.

The derivative of f with respect x can be defined by using first principle, expressed as;

$\frac{\partial f}{\partial x} =\lim_{h→0}\frac{f(x+h,y) – f(x,y)}{h}$

And the derivative of f with respect to y is,

$\frac{\partial f}{\partial y}=\lim_{h→0}\frac{f(x,y+h) – f(x,y)}{h}$

The partial derivative can be used for a function depending on three variables.

Partial Derivative Formula for three variables

If a function depends on three variables x, y and z, the derivative of this function can be calculated by using the following partial derivative formula. 

$f’[f(x,y,z)] =\frac{\partial f}{\partial x} +\frac{\partial f}{\partial y} +\frac{\partial f}{\partial z}$

Where, 

• $\frac{\partial f}{\partial x}$ is the derivative of f(x,y,z) with respect to x.
• $\frac{\partial f}{\partial y}$ is the derivative of f(x,y,z) with respect to y.
• $\frac{\partial f}{\partial z}$ is the derivative of f(x,y,z) with respect to z.

In calculus, the formula of a partial derivative can be modified to use it with other derivative rules. 

Partial Derivative and Product Rule

If there are two multivariable functions multiplied with each other, the product rule with the partial derivative is used to calculate rate of change. For example, the derivative of $F=f(x,y)g(x,y)$ can be calculated as;

$\frac{\partial F}{\partial x} = g(x,y)\frac{\partial f}{\partial x} + f(x,y)\frac{\partial g}{\partial x}$

And,

$\frac{\partial F}{\partial y} = g(x,y)\frac{\partial f}{\partial y} + f(x,y)\frac{\partial g}{\partial y}$

Partial Derivative and Power Rule

Since the partial derivative is used to calculate derivative of a multivariable function. It can be used along with the power rule if the function contains an algebraic function with degree n. The relation between power rule and partial differentiation for a function f(x,y) = xmyn is expressed as;

$f’(x,y) =\frac{\partial}{\partial x}(x^my^n) + \frac{\partial}{\partial y}(x^my^n)$

By using power rule formula,

$f’(x,y) = mx^{m-1}y^n + nx^my^{n-1}$

Where, the derivative of f(x,y) with respect to x is mxm-1 and the derivative of f(x,y) with respect to y is nxn-1.

Partial Derivative and Quotient Rule

If a multivariable function is divided by another function, the partial derivative along with the quotient rule to find the derivative. For a quotient of a function h(x,y) = f(x,y)/g(x,y), the relation between partial derivative and quotient rule is,

$\frac{\partial h}{\partial x} = \frac{g(x,y)\frac{\partial}{\partial x}[f(x,y)] – f(x,y)\frac{\partial}{\partial x}[g(x,y)]}{[g(x,y)]^2}$

And,

$\frac{\partial h}{\partial y} = \frac{g(x,y)\frac{\partial}{\partial y}[f(x,y)] – f(x,y)\frac{\partial}{\partial y}[g(x,y)]}{[g(x,y)]^2}$

By adding up, 

$h’(x,y) = \frac{\partial h}{\partial x} + \frac{\partial h}{\partial y}$

How to apply a partial derivative formula?

The implementation of partial derivative is divided into a few steps. These steps assist us to calculate the derivative of a function having more than one independent variables. These steps are:

1. Write the expression of the function. 
2. Identify the number of independent variables. 
3. Differentiate the function with respect to all independent variables one-by-one.
4. Find the sum of derivatives calculated in step 3 as $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$.
5. Simplify if needed. 

Let’s understand the following examples by applying partial derivative.

Partial Derivative Example 1

Suppose that a function f depends on two variables x and y which is written as,

$f= x^2 + 3xy$

To calculate derivative of this function, we will use the following steps,

$f(x,y) = x^2 +3xy$

Differentiating with respect to x, we get,

$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 +3xy) = 2x + 3y$

Now differentiating with respect to y, 

$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2 +3xy) = 3x$

Since the partial derivative formula is,

$f’(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$

Substituting the values of df/dx and df/dy, 

$f’(x,y) = 2x + 3y + 3x = 5x + 3y$

Hence the derivative of f= x2 + 3xy is 5x + 3y.

Partial derivative example 2

Suppose that a function f depends on three variables x, y and z which is written as,

$f= 2x^3yz^4 + 3x^2y^3z$

To calculate derivative of this function, we will use the following steps,

$f(x,y,z) = 2x^3yz^4 + 3x^2y^3z$

Differentiating with respect to x, we get,

$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (2x^3yz^4 + 3x^2y^3z) = 6x^2yz^4 + 6xy^3z$

Now differentiating with respect to y, 

$\frac{\partial f}{\partial y} = \frac{\partial f}{\partial y}(2x^3yz^4 + 3x^2y^3z) = 2x^3z^4 + 9x^2y^2z$

Again differentiating with respect to z,

$\frac{\partial f}{\partial z} = \frac{\partial f}{\partial z} (2x^3yz^4 + 3x^2y^3z) = 8x^3yz^3 + 3x^2y^3$

Since the partial derivative formula for three variables is,

$f’(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}

Substituting the values of df/dx and df/dy, 

$f’(x,y) = 6x^2yz^4 + 6xy^3z +2x^3z^4 + 9x^2y^2z + 8x^3yz^3 + 3x^2y^3$

Applying partial derivative formula by using calculator

The derivative of a multivariable function can be also calculated by using a partial derivative calculator. It is an online tool that follows the partial differentiation formula to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of $x^3y^3z^3$, the following steps are used by using this calculator.

1. Write the expression of the function in the input box such as, $x^3y^3z^3$.
2. Choose the variable to calculate the rate of change, which will be x, y and z in this example.
3. Review the input so that there will be no syntax error in the function. 
4. Now at the last step, click on the calculate button. By using this step, the partial derivative calculator will provide the derivative of $x^3y^3z^3$ quickly and accurately which will be $3(x^2y^3z^3 +x^3y^2z^3 + x^3y^3z^2)$.

Comparison between Partial derivative and implicit differentiation

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

Conclusion

Partial differentiation is a fundamental derivative concept used to find the rate of change of a function with respect to different variables. It can be used with other derivative rules depending on the function for which you want to calculate the derivative. It has many applications in calculus as well as other fields of science.