求解函数$f(x,y)=\frac{Ax + By}{Cx + Dy}$梯度时出错，请求排查问题

Hey there! Let's walk through where you might have stumbled when computing the gradient of this rational function. First, a quick recap: the gradient $\nabla f$ is just the vector of partial derivatives $\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$, so we need to nail each partial derivative using the quotient rule (since this is a ratio of two linear functions).

First, Let's Cover the Correct Partial Derivatives #

Let's define $u = Ax + By$ and $v = Cx + Dy$ for clarity. The quotient rule for derivatives is:
$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$$
When taking partial derivatives, we treat the non-target variable as a constant.

1. Partial derivative with respect to $x$ ($\frac{\partial f}{\partial x}$)

• $u_x = \frac{\partial u}{\partial x} = A$ (since $By$ is constant with respect to $x$)
• $v_x = \frac{\partial v}{\partial x} = C$ (since $Dy$ is constant with respect to $x$)
• Applying the quotient rule:
  $$\frac{\partial f}{\partial x} = \frac{A(Cx + Dy) - (Ax + By)C}{(Cx + Dy)^2}$$
• Expand and simplify the numerator:
  $$ACx + ADy - ACx - BCy = (AD - BC)y$$
• Final result:
  $$\frac{\partial f}{\partial x} = \frac{(AD - BC)y}{(Cx + Dy)^2}$$

2. Partial derivative with respect to $y$ ($\frac{\partial f}{\partial y}$)

• $u_y = \frac{\partial u}{\partial y} = B$ (since $Ax$ is constant with respect to $y$)
• $v_y = \frac{\partial v}{\partial y} = D$ (since $Cx$ is constant with respect to $y$)
• Applying the quotient rule:
  $$\frac{\partial f}{\partial y} = \frac{B(Cx + Dy) - (Ax + By)D}{(Cx + Dy)^2}$$
• Expand and simplify the numerator:
  $$BCx + BDy - ADx - BDy = (BC - AD)x = -(AD - BC)x$$
• Final result:
  $$\frac{\partial f}{\partial y} = \frac{-(AD - BC)x}{(Cx + Dy)^2}$$

Common Mistakes to Check For #

Here are the most frequent errors people make with this calculation:

• Mixed-up quotient rule order: Accidentally using $uv' - u'v$ instead of $u'v - uv'$. This flips all the signs in your numerator, leading to an incorrect gradient.
• Ignoring the constant variable: When taking $\frac{\partial f}{\partial x}$, forgetting that $By$ and $Dy$ are constants (so their partial derivatives with respect to $x$ are 0). Or vice versa for $\frac{\partial f}{\partial y}$.
• Skipping the denominator square: Forgetting to square $v = Cx + Dy$ in the quotient rule. This is a classic slip-up that makes your partial derivatives scale incorrectly.
• Algebra errors in simplification: Messing up the expansion of terms (e.g., $A(Cx + Dy)$) or failing to cancel out like terms (like $ACx$ and $-ACx$ in the $x$-partial derivative).

If you share your specific incorrect result, we can pinpoint exactly where you went off track, but these are the key spots to double-check!

内容的提问来源于stack exchange，提问作者Hritik