Partial Derivatives 3F4C43

1. **State the problem:** Find the first partial derivatives $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial y}$ of the function $$h(x,y) = (5 + 3y)^x$$. 2. **Recall the formula:** For a function of the form $$h(x,y) = f(y)^x$$, the partial derivatives are found using the chain rule and logarithmic differentiation. 3. **Partial derivative with respect to $x$:** Treat $y$ as constant. $$\frac{\partial h}{\partial x} = (5 + 3y)^x \ln(5 + 3y)$$ 4. **Partial derivative with respect to $y$:** Treat $x$ as constant. Use the chain rule: $$\frac{\partial h}{\partial y} = x (5 + 3y)^{x-1} \cdot \frac{\partial}{\partial y}(5 + 3y) = x (5 + 3y)^{x-1} \cdot 3$$ 5. **Final answers:** $$\frac{\partial h}{\partial x} = (5 + 3y)^x \ln(5 + 3y)$$ $$\frac{\partial h}{\partial y} = 3x (5 + 3y)^{x-1}$$