두 삼변수함수 $f, g$의 곱으로 표현된 삼변수함수 $h$의 순간변화율을 구해달라고 요청해봅시다.
[!user] User
Let $h(x, y, z)=f(x, y, z) g(x, y, z)$. Express the rate of change of $h$ at the point $P(x_0, y_0, z_0)$.
[!gpt] ChatGPT
To express the rate of change of $h$ at the point $P(x_0, y_0, z_0)$, we can compute the gradient of $h$. The gradient is a vector that contains all the first order partial derivatives of $h$.Let’s find the partial derivatives of $h$:
- $\frac{\partial h}{\partial x}$
- $\frac{\partial h}{\partial y}$
- $\frac{\partial h}{\partial z}$
Using the product rule for differentiation, for any two functions $u$ and $v$, we have:
\[\frac{d(uv)}{dx} = u'v + uv'\]\[\frac{\partial h}{\partial x} = \frac{\partial f}{\partial x} g + f \frac{\partial g}{\partial x}\]
- For $\frac{\partial h}{\partial x}$:
\[\frac{\partial h}{\partial y} = \frac{\partial f}{\partial y} g + f \frac{\partial g}{\partial y}\]
- For $\frac{\partial h}{\partial y}$:
\[\frac{\partial h}{\partial z} = \frac{\partial f}{\partial z} g + f \frac{\partial g}{\partial z}\]
- For $\frac{\partial h}{\partial z}$:
Evaluating these at the point $P(x_0, y_0, z_0)$, we get:
$\frac{\partial h}{\partial x} \Big _P = \frac{\partial f}{\partial x} (x_0, y_0, z_0) g(x_0, y_0, z_0) + f(x_0, y_0, z_0) \frac{\partial g}{\partial x} (x_0, y_0, z_0)$
$\frac{\partial h}{\partial y} \Big _P = \frac{\partial f}{\partial y} (x_0, y_0, z_0) g(x_0, y_0, z_0) + f(x_0, y_0, z_0) \frac{\partial g}{\partial y} (x_0, y_0, z_0)$
$\frac{\partial h}{\partial z} \Big _P = \frac{\partial f}{\partial z} (x_0, y_0, z_0) g(x_0, y_0, z_0) + f(x_0, y_0, z_0) \frac{\partial g}{\partial z} (x_0, y_0, z_0)$ Thus, the gradient of $h$ at point $P(x_0, y_0, z_0)$ is:
\[\nabla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y}, \frac{\partial h}{\partial z} \right) \Big|_P\]This gradient vector $\nabla h$ represents the rate of change of the function $h$ in the direction of maximum increase at the point $P$.