**Math Calculators** ▶ Partial Derivative Calculator

An online partial derivative calculator will determine the derivatives for the given function with many variables. This multivariable derivative calculator can differentiate the certain function multiple times. In the following guide, you can understand chain rule partial derivatives and much more.

In mathematics, the partial derivative of a multi-derivative function is defined as the derivative of a multi-variable function with respect to one variable, and all other variables remain unchanged.

When a function has two variables x and y that are independent of each other, then what to do there! Simply, if you require differentiating the function with respect to “x”, then you should keep the variable “y” constant and differentiate. On the other hand, if need to differentiate the function with respect to “y”, then make the variable “x” constant. The symbol “∂” is generally used to indicate partial derivatives.

You can use partial derivatives calculator to find the derivatives quickly. Otherwise, you can do these derivation calculations of a function manually by stick to these steps:

- Take a function to compute the partial derivative.
- The derivative of a constant is zero.
- When applying a derivative to a variable, only the derivative of that particular variable is solved.
- Solve all the functions for getting the results.

**Example**

**Question: Solve \( ∂^2/∂x [8x^2y^4+x^2] \) using partial differentiation method?**

**Solution:**

Given that,

$$ ∂^2/∂x [8x^2y^4+x^2]$$

$$ = ∂/∂x[∂/∂x(8x^2y^4 +x^2)] $$

Let us take function \( f=8x^2y^4+x^2 \), the partial differential calculatoruses the partial differentiation method for certain function:

$$ ∂f/∂x= 16xy^4 +2x $$

We can also write

$$ ∂^2/∂x [8x^2y^4+x^2] = ∂/∂x(∂f/∂x) $$

Now, the partial differentiation calculator applies the power rule on the function:

$$ =∂/∂x (16xy^4+2x) $$

Here, partial derivative calculator considers“y” as constant,

So,

$$ =16y^4 +2 $$

$$ ∂^2/∂x [8x^2y^4+x^2 = 16y^4+2 = 2(8y^4+1) $$

The high-order derivative is very important for testing the concavity of the function and confirming whether the endpoint of the function is maximum or minimum.

Since the function f (x, y) is continuously differentiable in the open region, you can obtain the following set of partial second-order derivatives:

**Direct partial second-order derivatives: **

F_{xx} = ∂fx / ∂x, where function f (x) is the first partial derivative of x.

f_{yy} = ∂fy / ∂y, where function f (y) is the first order derivative with respect to y.

**Cross partial derivative:**

fxy = ∂fx / ∂y, where f (x) is the first derivative with respect to x.

fyx = ∂fy / ∂x, where f (y) is the first partial derivative with respect to y.

However, an online Directional Derivative Calculator determines the directional derivative and gradient of a function at a given point of a vector.

**Example: **

**Calculate the first, second, and cross partial derivatives for the following function:**

**F (x, y) = x^2 + 10xy + 2y^2**

**Solution: **

**First order partial derivatives:**

Fx = 2x + 10y + 0 = 2x + 10y

Fy = 0 + 10x + 4y = 10x + 4y

Second order partial derivative calculator takes the second-order direct partial derivatives:

Fxx = ∂/∂x (2x + 10y) = 2

Fyy = ∂/∂y (10x + 4y) = 4

Second partial derivative calculator takes cross partial derivatives:

Fxy = ∂/∂y (2x + 10y) = 5

fyx=∂/∂x(10x+ 4y) = 5

Assume that x = g (a) and y = h (a) are differentiable functions of “a”, and z = f (x, y) are differentiable functions of x and y. Then z = f (x (a), y (a)) is a differentiable function of “a” and

Dz/da = ∂z/∂x⋅dx/da + ∂z/∂y⋅dy/da

The ordinary derivative is estimated at a, and the partial derivative is estimated at (x, y).

An online multivariable derivative calculator differentiates the given functions by taking the derivative by following these steps:

- First, enter a function for differentiation.
- Now, select the variable for derivative from the drop-down list.
- Then, select how many times you need to differentiate the given function.
- Hit the calculate button to see the results.

- The partial derivative calculator provides the derivative of the given function, then applies the power rule to obtain the partial derivative of the given function.

The chain rule says that the derivative f (g (x)) is equal to f'(g (x)) ⋅g’ (x). It helps us to differentiate the composite functions using the chain rule and the derivative of sin (x) and x^2, we can then determine the derivative of sin (x)^2.

Once you find the point where the gradient of the multivariable function is the zero vector, which means that the tangent plane of the graph is flat at that point, you can use the second-order partial derivative to determine whether the point is a local maxima, minima, or a saddle point.

An online Partial derivative calculator is used to differentiate mathematical functions that contain multiple variables. Therefore, partial differentiation is more general than ordinary differentiation. Partial differentiation is used to find the minima and maxima points in the optimization problem.

From the source of Wikipedia: Surface in Euclidean space, abuse of notation, Clairaut’s theorem, Optimization, Thermodynamics, quantum mechanics and mathematical physics.

From the source of Brilliant: instantaneous rate of change or slope, single-variable differentiation, Linearity, Product Rule, Chain Rule, Vector Calculus and Higher-order Derivatives, mixed derivative.

From the source of Khan Academy: multivariable function, three-dimensional graphs, single variable calculus, two-dimensional inputs, pre-evaluating.