Partial Derivative

The partial derivative is used in vector calculus and differential geometry. In Mathematics, sometimes the function depends on two or more variables. Here, the derivative converts into the partial derivative since the function depends on several variables. We will learn more about its definition, formula, rules, examples and other related topics.

Partial Derivative Definition

Suppose, we have a function f(x,y), which depends on two variables x and y, where x and y are independent of each other. Then we say that the function f partially depends on x and y. Now, if we calculate the derivative of f, then that derivative is known as the partial derivative of f. If we differentiate function f with respect to x, then take y as a constant and if we differentiate f with respect to y, then take x as a constant.

In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant.

For example, suppose f is a function in x and y then it will be expressed by f(x,y). So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. It should be noted that it is ∂x, not dx.

∂f/∂x is also known as fx.

Partial Derivative Symbol

The partial derivative of a function f with respect to the differently x is variously denoted by f’x,fx, ∂xf or ∂f/∂x. Here ∂ is the symbol of the partial derivative.

Partial Derivative of In

To find the partial derivative of natural logarithm “In”, we have to proceed with the same procedure as finding the derivative of the normal function. But, here when we calculate the partial derivative of the function with respect to one independent variable taking other as constant and follow the same thing with others.

Partial Derivative Formula

If f(x,y) is a function, where f partially depends on x and y and if we differentiate f with respect to x and y then the derivatives are called the partial derivative of f. The formula for partial derivative of f with respect to x taking y as a constant is given by;

Partial Derivative Formula

Partial Derivative Rules

Same as ordinary derivatives, partial derivatives follow some rule like product rule, quotient rule, chain rule etc.

Product Rule

If u = f(x,y).g(x,y), then,

Partial Derivative-Product Rule

Quotient Rule

If u = f(x,y)/g(x,y), where g(x,y) ≠ 0, then;

Partial Derivative-Quotient Rule

Power Rule

If u = [f(x,y)]2 then, the partial derivative of u with respect to x and y defined as;

ux = n|f(x,y)|n-1∂f/∂x

And uy = n|f(x,y)|n-1∂f/∂y

Chain Rule

If u = f(x,y) is a function where, x = (s,t) and y = (s,t) then by the chain rule, we can find the partial derivatives us and ut as:

Partial Derivative-Chain Rule


Problem: f(x,y) = x2y + sin x + cos y


Now, find out fx first keeping y as constant

fx = ∂f/∂x = (2x) y + cos x + 0

= 2xy + cos x

When we keep y as constant cos y becomes a constant so its derivative becomes zero.

Similarly, finding fy

fy = ∂f/∂y = x2 + 0 + (-sin y)

= x2 – sin y