Hessian Calculator
Find Hessian matrices step by step
The calculator will find the Hessian matrix of the multivariable function, with steps shown. Also, it will evaluate the Hessian at the given point if needed.
Your Input
Find the hessian matrix of the function $$$x^{3} + 4 x y^{2} + 5 y^{3} - 10$$$ with respect to $$$x$$$, $$$y$$$.
Solution
The entry at row $$$i$$$, column $$$j$$$ of the Hessian matrix is the partial derivative of the function with respect to the $$$i$$$-th and $$$j$$$-th variables.
$$$H_{11} = \frac{d^{2}}{dx^{2}} \left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right) = 6 x$$$ (for steps, see partial derivative calculator).
$$$H_{12} = \frac{d^{2}}{dydx} \left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right) = 8 y$$$ (for steps, see partial derivative calculator).
$$$H_{21} = \frac{d^{2}}{dxdy} \left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right) = 8 y$$$ (for steps, see partial derivative calculator).
$$$H_{22} = \frac{d^{2}}{dy^{2}} \left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right) = 2 \left(4 x + 15 y\right)$$$ (for steps, see partial derivative calculator).
Thus, $$$H = \left[\begin{array}{cc}6 x & 8 y\\8 y & 2 \left(4 x + 15 y\right)\end{array}\right]$$$.
Answer
$$$H = \left[\begin{array}{cc}6 x & 8 y\\8 y & 2 \left(4 x + 15 y\right)\end{array}\right]$$$A