What is Partial Derivative, Its Formula, Rules and Symbols

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. Questions in the CBSE board exams are often asked from this chapter.

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.

Following are the rules followed in partial derivatives:

1. 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

2. 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)

3. 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

4. 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. Even enrance exams like NEET and IISER cover questions on this topic.

Let us now understand the partial derivative symbol.

∂ is the universal symbol for “partial derivative.” It is also known as Glyph and spoken as “partial” or “partial dee” .

∂𝑓/∂𝑥 means “how f changes as we nudge x, holding everything else constant.” Without ∂, every multi-variable problem would remain undifferentiated. Why is regular d not used? Regular d (from 𝑑𝑦/𝑑x means “total change”. In simple words, every variable can freely change.

Curly ∂ ensures that all variables are in place except for one variable. First popularised in the late 1700s (Legendre/Euler era) as a stylised “d,” it is commonly known as the “curly d” or simply “del” (though true “del” ∇ is the gradient operator built from ∂’s).

Let us take a look at the key notations

First partial:

$$\frac{\partial f}{\partial x}$$

​Mixed partial:

$$\frac{{\partial }^{2}f}{\partial y\partial x}$$

Operator form:

Here are the three expressions :

1. First partial : $$\frac{\partial f}{\partial x}$$ 2. Mixed partial  $$\frac{{\partial }^{2}f}{\partial y\partial x}$$ 3. Operator form $$\frac{\partial }{\partial x}$$

Let us understand the partial derivative formula:

For a function f(x,y,…), the partial derivative with respect to x is defined by:

$$\frac{\partial f}{\partial x}(x,y,\dots )=\underset{h\to 0}{lim}\frac{f(x+h,y,\dots )-f(x,y,\dots )}{h}$$

Likewise, holding x (and any other variables) fixed, the partial derivative with respect to y is:

$$\frac{\partial f}{\partial y}(x,y,\dots )=\underset{k\to 0}{lim}\frac{f(x,y+k,\dots )-f(x,y,\dots )}{k}$$

Only one variable can be changed. Rest of the variables are fixed. The changing variable in the first formula will be x and in the second formula, it will be y. In IIT JAM and JEE Main, application-based questions related to the partial derivative formula will be asked from this topic. 

For those who plan to take entrance examinations after class 12th must practice questions based on partial derivatives. Let us consider some examples: 

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