Partial Derivative: Overview, Questions, Preparation

Mathematical functions can have two or more variables. When there are multiple variables, the partial derivative of the function is computed for each independent variable.

Partial Derivatives are an essential part of Calculus. 

In a function f (x,y), the partial derivative is calculated by differentiating f (x,y) separately with y as a constant and with x as a constant.

The formulae for these operations are given by

f_x = ꝺf / ꝺx = lim _h→0 [f(x+h, y) - f(x, y)] /h

f_y = ꝺf / ꝺy = lim _h→0 [f(x, y+h) - f(x, y)] /h

The symbol del, ꝺ denotes partial derivatives.

The partial derivative of f (x,y) with respect to x where y is taken as constant can be represented as f_x or ꝺf / ꝺx.

Rules of Partial Derivatives

Power Rule

When u = [f(x,y)]ⁿ 

ꝺu / ꝺx = n [f(x,y)]^n-1 ꝺf / ꝺx

Similarly,

ꝺu / ꝺy = n [f(x,y)]^n-1 ꝺf / ꝺy

Product Rule 

When u = f(x,y).g(x,y)

ꝺu / ꝺx = g(x,y) (ꝺf / ꝺx) + f(x,y) (ꝺu / ꝺx)

Similarly,

ꝺu / ꝺy = g(x,y) (ꝺf / ꝺy) + f(x,y) (ꝺu / ꝺy)

Quotient Rule

When u = f(x,y)/g(x,y)   (where g(x,y) ≠ 0 )

ꝺu / ꝺx =  [g(x,y) (ꝺf / ꝺx) + f(x,y) (ꝺu / ꝺx)] /  [g(x,y)]²

Similarly,

ꝺu / ꝺy =  [g(x,y) (ꝺf / ꝺy) + f(x,y) (ꝺu / ꝺy)] /  [g(x,y)]²

Chain Rule

-For one independent variable:

If x = g(t) and  y =h(t) and  z = f(x, y)

z =  f(g(t), h(t))

Partial derivative with respect to t is,

ꝺz / ꝺt = (ꝺz / ꝺx) (ꝺx / ꝺt) + (ꝺz / ꝺy) (ꝺy / ꝺt)

Partial derivatives are a part of multivariate Calculus and are taught at advanced Mathematical levels. The basic differentiation rules for these operations are learnt in class 11 and 12. This subject is used often in Science and Mathematics.

1. Find the partial derivative of the function: f (x,y) = 12x - 4y.

Solution.

ꝺf / ꝺx = 12   (differentiation of f (x,y) with y constant)

Similarly,

ꝺf / ꝺy = -4

2.Find the partial derivative of z = (2x+y) (x-y)

Solution.

This function is in the format u = f(x,y).g(x,y)

Using Product Rule,

ꝺz / ꝺx = g(x,y) (ꝺf / ꝺx) + f(x,y) (ꝺu / ꝺx)

Where 

g(x,y) = (x-y)

 f(x,y) = (2x+y)

ꝺz / ꝺx = (x-y) (2) + (2x+y) (-1)

=2x -2y -2x -y

= -3y

3.Find the partial derivative of z = (2x +y²)³

Solution.

This function is in the format u = [f(x,y)]ⁿ  

Using power rule, 

ꝺz / ꝺx =  n [f(x,y)]^n-1 ꝺf / ꝺy 

Where f(x,y) = 2x +y²  and n=3

ꝺz / ꝺx =3 (2x +y²)² (2)

= 6 (2x +y²)²

Q: When are partial derivatives used?

Q: How do you find the Partial Derivative of a Natural Logarithm (In)?

Q: What are the rules to solve Partial Derivatives?

Q: Can partial derivatives be used for higher-order equations?

Q: How do you use the chain rule for two independent variables?