The equation in the image is:

$z = \frac{x}{x + y}$

We need to compute the derivative of $z$ with respect to $x$, assuming $y$ is a constant. This is a quotient rule derivative problem, as the function $z$ is a ratio of two functions of $x$.

The quotient rule for derivatives is given by:

$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$

Here,

• $u = x$
• $v = x + y$

Now, let's compute the derivative step by step:

$\frac{dz}{dx} = \frac{(x + y) \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(x + y)}{(x + y)^2}$

Since $\frac{d}{dx}(x) = 1$ and $\frac{d}{dx}(x + y) = 1$ (as $y$ is constant), the derivative simplifies to:

$\frac{dz}{dx} = \frac{(x + y) \cdot 1 - x \cdot 1}{(x + y)^2}$

Simplifying further:

$\frac{dz}{dx} = \frac{x + y - x}{(x + y)^2} = \frac{y}{(x + y)^2}$

Thus, the derivative of $z$ with respect to $x$ is:

$\frac{dz}{dx} = \frac{y}{(x + y)^2}$

Would you like more details on this, or do you have any specific questions?

Here are 5 related questions for further exploration:

1. How would the derivative change if $y$ were also a function of $x$?
2. What happens to the derivative as $y \to 0$?
3. Can you apply the chain rule to compute a second derivative of $z$ with respect to $x$?
4. How does the quotient rule compare with the product rule in solving such derivatives?
5. Can you find the partial derivative of $z$ with respect to $y$?

Tip: The quotient rule is essential when differentiating functions that are ratios of two differentiable functions. Always identify $u$ and $v$ clearly.