$$ \frac{\partial z}{\partial x} $$, where $$ x y-z=4 $$ $$ \frac{\partial z}{\partial x}=\square $$

Question

UpStudy AI · Accepted Answer

To find $$ \frac{\partial z}{\partial x} $$ from the equation $$ x y - z = 4 $$, we can use implicit differentiation.

1. Start with the given equation:
   $$
   x y - z = 4
   $$

2. Differentiate both sides with respect to $$ x $$:
   - The derivative of $$ x y $$ with respect to $$ x $$ is $$ y + x \frac{\partial y}{\partial x} $$ (using the product rule).
   - The derivative of $$ z $$ with respect to $$ x $$ is $$ \frac{\partial z}{\partial x} $$.
   - The derivative of the constant $$ 4 $$ is $$ 0 $$.

Thus, we have:
   $$
   \frac{d}{dx}(x y) - \frac{\partial z}{\partial x} = 0
   $$
   which simplifies to:
   $$
   y + x \frac{\partial y}{\partial x} - \frac{\partial z}{\partial x} = 0
   $$

3. Rearranging gives:
   $$
   \frac{\partial z}{\partial x} = y + x \frac{\partial y}{\partial x}
   $$

Now, we can express $$ \frac{\partial z}{\partial x} $$ in terms of $$ y $$ and $$ \frac{\partial y}{\partial x} $$.

If we assume $$ y $$ is a function of $$ x $$, we can leave it in this form. If $$ y $$ is constant, then $$ \frac{\partial y}{\partial x} = 0 $$, and we have:
$$
\frac{\partial z}{\partial x} = y
$$

Thus, the final expression for $$ \frac{\partial z}{\partial x} $$ is:
$$
\frac{\partial z}{\partial x} = y + x \frac{\partial y}{\partial x}
$$

If you have specific values for $$ y $$ or $$ \frac{\partial y}{\partial x} $$, please provide them for a numerical answer.
$$ \frac{\partial z}{\partial x} = y + x \frac{\partial y}{\partial x} $$

\( \frac{\partial z}{\partial x} \), where \( x y-z=4 \) \( \frac{\partial z}{\partial x}=\square \)

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions