Education

# How Do You Find the Partial Derivative of a Function?

## Find the Partial Derivative

How Do You Find the Partial Derivative of a Function?

The partial derivatives are used in vector calculus and differential geometry to find derivatives of a
function having several, but we find the derivative with respect to one of its variables. By using the partial
derivatives we find the differential equation, which is used to find the points of maximum and the
minimum points when analyzing a surface, or plotting a graph. We can use the partial derivatives
calculator to find the partial derivative of the function in a three-dimensional plane.  Basically, we do
derivation to find the rate of change or a slope of a tangent line. When we are working on the three-
dimensional surface, we use the partial derivative to find how the function is changing at any given point;
these surfaces are perpendicular to each other, in a three-dimensional plane.
In this article we try to understand the partial derivative and the simple method of how to solve it:

## How we can define the partial derivative

We can understand the partial derivative by its definition, considering a function f(x,y), the function depends
on two variables x and y, the x and y are independent of each other but have a direct relation with the rate
of change of the function f. In this situation, the derivative we get is a partial derivative. When we solve
the derivative with respect to x, then we consider y is a constant if we are calculating the derivative with
respect to y, we then consider x and a constant one. The partial derivative can be calculated by a partial
derivative solver, this would also increase your understanding of the partial derivative.

## The partial derivative symbol and formula.

When we are calculating the partial derivative with respect to x, then the symbols of the partial derivative
with respect to the x are f’x, ∂xf or ∂f/∂x, the symbol of the partial derivative is “∂”.
Now if f(x,y) is a function, and we want to find the partial derivative with respect to x and keep the “y” as
a constant, Then the  formula would be:
fx=fx=limh0f(x+h,y)-f(x,y)h
Now the formula for the partial derivative with respect to y, and keeping the x a constant, we get:
fy=fy=limh0f(x,y+h)-f(x,y)h
We can find the second derivative by using the partial derivatives calculator, for our convenience.

## Example:

Now  consider a function f =  x 3 + 4xy, this function has more than one variable, we can find its derivative
by partial derivative method, we can use the partial derivative calculator to find the partial derivative of this
function:
fx=x( x 3 + 4xy)=3×2+4y
Now to find the tangent line slope at point P(2,2) and we can put the values to find the slope of the tangent:
The value of the fxat (2,2) is :
3×2+4y=3(2)2+ 4(2)=3(4)+8=12+8=20

This means the slope of the tangent line is 20, therefore the x=20 at (2,2), for finding the partial derivative
directly, we can use the partial derivative calculator.

## Partial derivative rules:

There are four types of rules for partial derivative, product rule, quotient rule, power, and the chain rule for
partial derivative:
Product rule:
If  we need to multiply two partial derivatives, we apply the following product rule:
If u=f(x,y).g(x,y) , then we multiply as:
Ux=Ux=g(x,y) fx+f(x,y) gx
Uy=Uy=g(x,y) fy+f(x,y) gy
The product of derivatives can be found by using the partial derivative calculator, to avoid the length
product calculations.

There are separate formulas for quotient, power, and chain rules formulas for the partial derivative; we can
use the derivative chain rule calculator to find the partial derivative of the independent variables.