The partial derivative is the derivative of an equation, with more than one variable, with respect to one of those variables. For example you may take the partial derivative of 2x+3y with respect to x. The partial derivative of a function with respect to x may be represented by ∂f/∂x. Note the difference between the basic derivative df/dx and the partial derivative ∂f/∂x.
Step2
For finding the partial derivative with respect to x of the equation 2x+3y, we would treat the x as a variable and treat the y as a constant. Since x is being treated as a variable, the derivative of 2x would be 2 using basic derivatives. Since y is being treated as a constant, the derivative of 3y would be 0. Therefore, the derivative of 2x+3y would be 2+0, or simply 2.
Step3
When we are taking the partial derivative we treat the other "letters" in the equation the same as we would treat a number. For example, using the above problem 2x+3y, when we take the partial derivative with respect to x, we treat y like a constant. This means we would treat y and therefore 3y as a number. Therefore, the derivative of 3y is the same as the derivative of a number, or 0.
Step4
Now to move on to a more complex equation. We will be taking the partial derivative with respect to y of the equation x^2+xy+y^2. Since we are now taking the derivative with respect to y, we will be treating x as a constant and treating y as a variable. Since x is a constant, then x^2 would also be a constant and the derivative would be 0. For the derivative of xy, it would be similar taking the basic derivative df/dy of 2y, where 2 would be constant and y would be the variable. df/dy of 2y would then be 2. Since for our problem x is constant and y is a variable, we would get x. Finally for the derivative of y^2, since y is a variable, we would get 2y. Therefore our answer is 0+x+2y or simply x+2y.
Step5
For the final example, I will be taking the partial derivative of x*y^2+y*x^2 with respect to x. We are treating x as a variable and y as a contant. The derivative of x*y^2 would be x. The derivative of y*x^2 would be y times the derivative of x^2 or y*2x or simply 2xy. Therefore the overall derivative would be x+2xy.
Tips & Warnings
When treating a "letter" like a constant, it is the same as treating it like a number.
When treating a "letter" like a variable, you treat it the same as you would when solving for the basic derivative.
Enthusiast 
