# Second Order Partial Derivative Calculator

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This is a second order partial derivative calculator.

A partial derivative is a derivative taken of a function with respect to a specific variable.

The function is a multivariate function, which normally contains 2 variables, x and y. However, the function may contain more than 2 variables.

So when we take the partial derivative of a function, we take it with respect to a specific variable.

A second order derivative is the second derivative of a function. A second order derivative takes the derivative to the 2nd order, which is really taking the derivative of a function twice.

A second order partial derivative is simply a partial derivative taken to a second order with respect to the variable you are differentiating to.

As an example, let's say we want to take the partial derivative
of the function, f(x)= x^{3}y^{5}, with respect to x, to the 2nd order. This is represented by ∂^{2}f/∂x^{2}.

So since we are finding the derivative with respect to x, we find
the derivative of the x component of the function. Since x is raised to the power of 3, the derivative of
the first order of the x component is 3x^{2}. Taking the derivative of 3x^{2} gives us 6x. This is gotten simply by using the power rule
in calculcus.

Since we are not taking the derivative of the function with respect to y, we leave the y component unchanged.

Thus, the complete partial derivative of the function,
x^{3}y^{5}, with respect to x, to the second order, is 6xy^{5}

Now let's do the same function but now find the partial derivative of it with respect to y.

So, again, the original function is, (x)= x^{3}y^{5}

Now we are simply going to find the partial derivative with respect to y to the second order.

So, again, using the power rule in calculus, we can find the
derivative of the y component of the function. Doing so gives us, 5y^{4}. Taking the derivative of
5y^{4} gives us 20y^{3}.

The x component of the function is unchanged, because we are not finding the derivative of the function with respect to x.

Thus, the partial derivative of the function,
x^{3}y^{5}, with respect to y, is 20x^{3}y^{3}

Partial differentiation is important when you want to see how the rate of change of one variable affects a function that has multiple variables. By taking the partial derivative of a function, we can see how the rate of change of that one variable affects the entire function.

Normally, partial differentiation is done on functions that contain 2 variables, but some functions may have more.

From a technical perspective, for those who want to know the technical aspect of it, this calculator is built using the sympy module in the Python programming language.

This calculator can take the partial derivative of regular functions, as well as trigonometric functions.

This user simply enters in the function, the variable to
differentiate with respect to, and the higher order of the derivative of which to calculate to. The resultant partial derivative will then be
automatically computed and displayed.

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