Problem 11 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the following functions. $$f(x, y)=x e^{y}$$

Step-by-Step Solution

Answer

Question: Find the first partial derivatives of the function f(x, y) = x * e^y with respect to x and y. Answer: The first partial derivatives are: $$\frac{\partial f}{\partial x} = e^{y}$$ $$\frac{\partial f}{\partial y} = x e^{y}$$

Step 1: Identify the given function

We are given the function: $$f(x, y) = x e^{y}$$

Step 2: Find the partial derivative with respect to x

Keep y constant and take the derivative of f(x, y) with respect to x: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x e^{y})$$ Y is treated as a constant in this derivative so we just differentiate x: $$\frac{\partial f}{\partial x} = e^{y}$$

Step 3: Find the partial derivative with respect to y

Now, keep x constant and take the derivative of f(x, y) with respect to y: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x e^{y})$$ X is treated as a constant in this derivative so we focus on differentiating e^y, and using the chain rule we have: $$\frac{\partial f}{\partial y} = x \frac{\partial}{\partial y}(e^{y})$$ $$\frac{\partial f}{\partial y} = x e^{y}$$

Step 4: Write the final solution

The first partial derivatives of the function f(x, y) = x * e^y with respect to x and y are: $$\frac{\partial f}{\partial x} = e^{y}$$ $$\frac{\partial f}{\partial y} = x e^{y}$$