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関数の勾配計算機

関数の勾配ベクトルを段階的に求める

この計算機は、与えられた関数の勾配ベクトル（必要に応じて指定された点で）を、計算過程を示しながら求めます。

Enter a function:
Enter the order of variables and/or a point:
If you don't need the order of variables, leave it empty.
If you want a specific order of variables, enter variables comma-separated, like `x,y,z`.
If you want the gradient at a specific point, for example, at `(1, 2, 3)`, enter it as `x,y,z=1,2,3`, or simply `1,2,3` if you want the order of variables to be detected automatically.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: find the gradient of $$$f=x^{3} + y^{5}$$$at $$$\left(x,y\right)=\left(1,7\right)$$$

To find the gradient of a function (which is a vector), differentiate the function with respect to each variable.

$$$\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$$$

$$$\frac{\partial f}{\partial x}=3 x^{2}$$$ (for steps, see derivative calculator)

$$$\frac{\partial f}{\partial y}=5 y^{4}$$$ (for steps, see derivative calculator)

Finally, plug in the point:

$$$\nabla f \left(1,7\right)=\left(3,12005\right)$$$

Answer

$$$\nabla \left(x^{3} + y^{5}\right) \left(x,y\right)=\left(3 x^{2},5 y^{4}\right)$$$

$$$\nabla \left(x^{3} + y^{5}\right)|_{\left(x,y\right)=\left(1,7\right)}=\left(3,12005\right)$$$