Partial Derivative of functions is an important topic in Calculus. If we have a function f(x,y) i.e. a function which depends on two variables x and y, where x and y are independent to each other, then we say that the function f partially depends on x and y. The derivative of f is called the partial derivative of f. When we differentiate f with respect to x, then consider y as a constant and when we differentiate f with respect to y, then consider x as a constant.
For example, suppose f is a function in x and y then it will be denoted by f(x,y). So, partial derivative of f with respect to x will be ∂x∂f keeping y terms as constant. Note that, it is not dx, instead it is ∂x.
It can also be denoted as f’x, fx, ∂xf or ∂x∂f
If f(x,y) is a function, where f is partially depends on x and y. Then if we differentiate f withe respect to x and y then the derivatives are called the partial derivative of f with respect to x and y. The formula for partial derivative of f with respect to x taking y as a constant,
fx = ∂x∂f = limh→0hf(x+h,y)–f(x,y)
And partial derivative of f with respect to y taking x as a constant,
fy = ∂y∂f = limh→0hf(x,y+h)–f(x,y)
Partial Derivative Formulas and Identities
There are some identities for partial derivatives as per the definition of the function.
1. If u = f(x,y) and both x and y are differentiable of t i.e. x = g(t) and y = h(t), then the term differentiation becomes total differentiation.
2. The total partial derivative of u with respect to t is dtdf=∂x∂fdtdx+∂y∂fdtdy
3. If f is a function defined as f(x), where x(u,v) then ∂u∂f=∂x∂f∂u∂x and ∂v∂f=∂x∂f∂v∂x
Suppose f = f(x,y) and y is a implicit function it means y is itself a function of x, then dxdf=∂x∂f+∂y∂fdxdy
4. If f(x,y) where x(u,v) and y(u,v), then ∂u∂f=∂x∂f∂u∂x+∂y∂f∂u∂y and ∂v∂f=∂x∂f∂v∂x+∂y∂f∂v∂y
First Partial Derivative
If u = f(x,y) is then the partial derivative of f with respect to x defined as ∂x∂f and denoted by
∂x∂f = limδx→0δxf(x+δx,y)−f(x,y)
And, partial derivative of f with respect to y defined as ∂y∂f and denoted by
∂y∂f = limδy→0δyf(x,y+δy)−f(x,y)
These are called the first partial derivatives of f. When we calculate the partial derivatives of f with respect to x treating y as a constant and vise versa.
Double Partial Derivative
Since the second order partial derivative can be found by differentiating the first partial derivative, we can also call it as the double partial derivative. Second order partial derivatives can be defined as follows:
∂x∂(∂x∂f) is denoted by ∂2x∂2f or fxx or fx2∂y∂(∂y∂f) is denoted by ∂y2x∂2f or fyy or fy2∂x∂(∂y∂f) is denoted by ∂x.∂y∂2f or fxy∂y∂(∂x∂f) is denoted by ∂y.∂x∂2f or fyx
The partial differentiation fxy and fyx are distinguished by the order on which ‘f’ is successively differentiated with respect to ‘x’ and ‘y’. In general, the two partial derivatives fxy and fyx need not be equal.
Second Partial Derivative Test
The necessary condition for the existence of relative maximum and relative minimum of a function of two variables f(x,y) is
∂x∂f = 0 and ∂y∂f = 0 ………………(a)
If (x1 , y1) are the points of the function which satisfying equation (a), then
If fxx < 0 and fyy < 0 then (x1 , y1) is the relative maximum point of the function.
If fxx < 0 and fyy < 0 then (x1 , y1) is the relative minimum point of the function.
Mixed Partial Derivative
We can find out the mixed partial derivative or cross partial derivative of any function when the second order partial derivative exists. If f(x,y) is a function of with two independent variables, then we know that
∂x∂f = limδx→0δxf(x+δx,y)–f(x,y) is the first partial derivative of f with respect to x.
The terms fxy and fyx are called the mixed or cross partial derivative of f.
Higher-Order Partial Derivatives
Second and higher-order partial derivatives are defined analogously to the higher-order derivatives of univariate functions. For the function f (x, y, . . .) the “own” second partial derivative with respect to x is simply the partial derivative of the partial derivative (both with respect to x).
The four second-order partial derivatives are as follows.
Let f (x, y) be a function of two variables such that ∂x∂f,∂y∂f both exist.
1] The partial derivative of ∂y∂f w.r.t. ′x′ is denoted by ∂x2∂2forfxx.
2] The partial derivative of ∂y∂f w.r.t. ′y′ is denoted by ∂y2∂2forfyy.
3] The partial derivative of ∂x∂f w.r.t. ′y′ is denoted by ∂y∂x∂2forfxy.
4] The partial derivative of ∂y∂f w.r.t. x is denoted by ∂y∂x∂2forfyx.