Partial Derivative

The partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants.

Let us learn more about how to calculate partial derivatives of different orders along with examples.

The partial derivative of a multivariable function, say z = f(x, y), is its derivative with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example,

• for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f / ∂x), y is treated as constant and
• for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f / ∂y), x is treated as constant

Note that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name "partial". The limit definition of a partial derivative looks very similar to the limit definition of the derivative. We can find the partial derivatives using the following limit formulas:

• ∂f / ∂x = lim _{h → 0} [ f(x + h, y) - f(x, y) ] / h
• ∂f / ∂y = lim _{h → 0} [ f(x, y + h) - f(x, y) ] / h

These formulas resemble the derivative definition using the first principle.

Example of Partial Derivative

If f (x, y) = xy, then find the partial derivative ∂f / ∂x.

Solution:

∂f / ∂x = lim _{h → 0} [ f(x + h, y) - f(x, y) ] / h
= lim _{h → 0} [ (x + h)y - xy ] / h
= lim _{h → 0} [xy + hy - xy] / h
= lim _{h → 0} [hy]/h
= lim _{h → 0} y
= y

Therefore, ∂f / ∂x = y.

We know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy/dx, d/dx (y), d/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol "∂" instead of "d". We pronounce "∂" to be "doh" but it has some other names like "partial", "del", "partial dee", "dee", "Jacobi's delta", etc. If z = f(x, y) is a function in two variables then

• ∂f / ∂x is the partial derivative of f with respect to x
• ∂f / ∂y is the partial derivative of f with respect to y

Just like how we have different symbols of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f / ∂x can be written as f_x, f_x', D_xf, ∂ / ∂x (f), ∂_x f, ∂ / ∂x [f(x, y)], ∂z / ∂x, etc.

We have already seen that the limit definitions are used to find the partial derivatives. But using the limit formula and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps:

• Step 1: Identify the variable with respect to which we have to find the partial derivative.
• Step 2: Except for the variable found in Step 1, treat all the other variables as constants.
• Step 3: Differentiate the function just using the rules of ordinary differentiation.

Wait! Read Step 3 again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it!

Example: Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f / ∂x) using the above steps.

Solution:

We have to find ∂f / ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d/dx (c y) = c dy/dx, where 'c' is a constant). Thus,

∂f / ∂x = ∂ / ∂x (xy)
= y ∂ / ∂x (x)
= y (1) (Using power rule, d/dx (x) = 1)
= y

We have got the same answer as we got using the limit definition.

We have derivatives like first-order derivatives (like dy/dx), second-order derivatives (like d²y/dx²), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also.

First Order Partial Derivatives

If z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f / ∂x and ∂f / ∂y.

Example: If z = x² + y², find all the first order partial derivatives.

Solution:

f_x = ∂f / ∂x = ∂ / ∂x (x² + y²)
= ∂ / ∂x (x²) + ∂ / ∂x (y²)
= 2x + 0 (as y is a constant)
= 2x

f_y = ∂f / ∂y = ∂ / ∂y (x² + y²)
= ∂ / ∂y (x²) + ∂ / ∂y (y²)
= 0 + 2y (as x is a constant)
= 2y

Second Order Partial Derivatives

The second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂²f / ∂x², ∂²f / ∂y², ∂²f / ∂x ∂y and ∂²f / ∂y ∂​x​​​. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e.,

• f_xx = ∂²f / ∂x² = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (f_x)
• f_yy = ∂²f / ∂y² = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (f_y)
• f_yx = ∂²f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (f_y)
• f_xy = ∂²f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (f_x)

Observe the notations f_yx and f_xy. The order of variables in each subscript indicate the order of partial differentiation. For example, f_yx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂²f / ∂x ∂y.

Example: If z = x² + y², find all the second order partial derivatives.

Solution:

In the above example, we have already found that f_x = 2x and f_y = 2y.

Now, f_xx = ∂ / ∂x (f_x) = ∂ / ∂x (2x) = 2

f_yy = ∂ / ∂y (f_y) = ∂ / ∂y (2y) = 2

f_yx = ∂ / ∂x (f_y) = ∂ / ∂x (2y) = 0

f_xy = ∂ / ∂y (f_x) = ∂ / ∂y (2x) = 0

Now that f_yx = f_xy. Thus, the order of partial differentiation doesn't matter.

The process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas:

• ∂f / ∂x = lim _{h → 0} [ f(x + h, y) - f(x, y) ] / h
• ∂f / ∂y = lim _{h → 0} [ f(x, y + h) - f(x, y) ] / h

But instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same rules as ordinary differentiation and among them, the important rules are as follows:

Power Rule

The power rule of differentiation says d/dx (xⁿ) = n x^n-1. The same rule can be applied in partial derivatives also.

Example: ∂ / ∂x (x²y) = y ∂ / ∂x (x²) = y (2x) = 2xy.

Product Rule

The product rule of ordinary differentiation says d/dx (uv) = u dv/dx + v du/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable.

Example: ∂ / ∂x (xy sin x) = y ∂ / ∂x (x sin x)
= y [ x ∂ / ∂x (sin x) + sin x ∂ / ∂x (x) ]
= y [ x cos x + sin x]

Quotient Rule

The quotient rule of ordinary differentiation says d/dx (u/v) = [ v du/dx - u dv/dx ] / v². As other rules, this rule can be applied for finding partial derivatives also.

Example: ∂ / ∂x ( xy / sin x)
= y ∂ / ∂x (x / sin x)
= y [ ( sin x ∂ / ∂x (x) - x ∂ / ∂x (sin x) ) / sin²x ]
= y [ sin x - x cos x] / sin²x

Chain Rule of Partial Differentiation

The chain rule is used when we have to differentiate an implicit function. The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined.

• If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then
  ∂f/∂u = ∂f/∂x · ∂x/∂u;
  ∂f/∂v = ∂f/∂x · ∂x/∂v

• If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then
  df/dt = (∂f/∂x · dx/dt) + (∂f/∂y · dy/dt)

• If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then
  ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u;
  ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v

Example: If z = e^xy, where x = uv and y = u + v then find the partial derivative ∂f/∂u.

Solution:

By the chain rule of partial derivatives:

∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u
= ∂ / ∂x (e^xy) · ∂ / ∂u (uv) + ∂ / ∂y (e^xy) · ∂ / ∂u (u + v)
= (e^xy · y) (v) + (e^xy · x) (1)
= e^xy (x + vy)

Other Rules of Partial Differentiation

• If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from implicit differentiation.
  dy/dx = -f_x/f_y.
• For any two functions u(x, y) and v(x, y), the determinant \(\left|\begin{array}{ll}
  \partial u / \partial x & \partial u / \partial y \\
  \partial v / \partial x & \partial v / \partial y
  \end{array}\right|\) is known as Jacobian of u and v.
• The Laplace equation of partial derivatives is ∂²f / ∂x² + ∂²f / ∂y² + ∂²f / ∂z² = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function.

Important Notes on Partial Derivatives:

• While finding the partial derivative with respect to a variable, all the other variables should be considered as constants.
• The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂²f / ∂x ∂y = ∂²f / ∂y ∂x.
• The rules of derivatives apply for partial differentiation as well.
• Instead of using the limit definition, applying derivative formulas make the process of finding the partial derivatives easier.

☛ Related Topics:

• Partial Differential Equations
• Partial Derivative Calculator

How to Calculate Partial Derivatives?

We use partial derivatives when the function has more than one variable. If a function f is in terms of two variables x and y, then we can calculate the partial derivatives as follows.

• the partial derivative of f = ∂f/∂x and y has to be treated as constant here.
• the partial derivative of f = ∂f/∂y and x has to be treated as constant here.

What is the Symbol of Partial Derivatives?

We use the symbol ∂ to represent a partial derivative. For example, the partial derivative of a function f(x, y) with respect to x is written as ∂f/∂x.

What is the Difference Between Differentiation and Partial Differentiation?

We talk about the derivative of a function if a function has only one variable in it. For example, the derivative of a function y = f(x) is denoted by df/dx. We talk about partial derivatives when a function z = f(x, y) has more than one variable. The partial derivative of f with respect to x is denoted by ∂f/∂x and while finding this, we treat y as constant.

What is the Formula Used to Find the Partial Derivative?

The partial derivatives of a function z = f(x, y) can be found using the limit formulas:

• ∂f / ∂x = lim _{h → 0} [ f(x + h, y) - f(x, y) ] / h
• ∂f / ∂y = lim _{h → 0} [ f(x, y + h) - f(x, y) ] / h

What Does Partial Derivative Tell Us?

Partial derivative tells us to differentiate a function partially. It means if we are differentiating partially with respect to one variable, then the remaining variables of the function must be treated as constants.

What is the Chain Rule of Partial Derivatives?

The chain rule of partial derivative is mentioned below: If z = f(x, y) is a function where x and y are functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then by the chain rule of partial derivatives,

• ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u
• ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v

What is the Formula that Connects Normal Derivatives with Partial Derivatives?

When f(x, y) = c, where 'c' is a constant, then dy/dx = -f_x/f_y, where

• f_x is the partial derivative of f with respect to x
• f_y is the partial derivative of f with respect to y