Let's solve the given problem step-by-step.

Problem Statement:

You are given a function of two variables:
$h(x, y) = (y^2 + 1) e^x$ The question asks you to compute the sum of the partial derivatives: $h_x(x, y) + h_y(x, y).$

Step 1: Find the partial derivative $h_x(x, y)$ (with respect to $x$).

Treat $y$ as a constant when differentiating with respect to $x$. Using the product rule on $(y^2 + 1) e^x$: $h_x(x, y) = (y^2 + 1) \cdot \frac{\partial}{\partial x}(e^x) = (y^2 + 1) e^x.$

Step 2: Find the partial derivative $h_y(x, y)$ (with respect to $y$).

Treat $x$ as a constant when differentiating with respect to $y$. The function is: $h(x, y) = (y^2 + 1) e^x.$ Since $e^x$ is treated as a constant here, we only need to differentiate $y^2 + 1$ with respect to $y$: $\frac{\partial}{\partial y}(y^2 + 1) = 2y.$ Thus: $h_y(x, y) = 2y \cdot e^x.$

Step 3: Sum the partial derivatives $h_x(x, y) + h_y(x, y)$.

Now, add the two partial derivatives together: $h_x(x, y) + h_y(x, y) = (y^2 + 1) e^x + 2y e^x.$ Factor out $e^x$: $h_x(x, y) + h_y(x, y) = e^x (y^2 + 2y + 1).$ Notice that: $y^2 + 2y + 1 = (y + 1)^2.$

Step 4: Find the correct answer.

Thus, the sum of the partial derivatives is: $h_x(x, y) + h_y(x, y) = e^x (y + 1)^2.$

Now, let’s compare with the given answer choices.

• Option (b): $e^x (y^2 + 2y + 1)$ is equivalent to our result.
  So, (b) is the correct answer.

Final Answer:

The correct option is (b).

Do you need further clarification or details on this solution?

Here are five related questions:

1. What is the meaning of partial derivatives in multivariable functions?
2. How does the product rule apply in partial differentiation?
3. Can you explain how to interpret the gradient of a multivariable function?
4. What is the difference between $h_x$ and $h_y$ geometrically?
5. How does factoring help simplify expressions in partial derivatives?

Tip: When differentiating products in partial derivatives, treat the variables independently to avoid confusion.