Calculadora de gradiente de função
Encontrar gradiente de função passo a passo
A calculadora encontrará o gradiente da função dada (no ponto dado, se necessário), com as etapas mostradas.
Solution
Your input: find the gradient of $$$f=e^{x} + \sin{\left(y z \right)}$$$at $$$\left(x,y,z\right)=\left(3,0,\frac{\pi}{3}\right)$$$
To find the gradient of a function (which is a vector), differentiate the function with respect to each variable.
$$$\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$$$
$$$\frac{\partial f}{\partial x}=e^{x}$$$ (for steps, see derivative calculator)
$$$\frac{\partial f}{\partial y}=z \cos{\left(y z \right)}$$$ (for steps, see derivative calculator)
$$$\frac{\partial f}{\partial z}=y \cos{\left(y z \right)}$$$ (for steps, see derivative calculator)
Finally, plug in the point:
$$$\nabla f \left(3,0,\frac{\pi}{3}\right)=\left(e^{3},\frac{\pi}{3},0\right)$$$
Answer
$$$\nabla \left(e^{x} + \sin{\left(y z \right)}\right) \left(x,y,z\right)=\left(e^{x},z \cos{\left(y z \right)},y \cos{\left(y z \right)}\right)$$$
$$$\nabla \left(e^{x} + \sin{\left(y z \right)}\right)|_{\left(x,y,z\right)=\left(3,0,\frac{\pi}{3}\right)}=\left(e^{3},\frac{\pi}{3},0\right)$$$