1. **State the problem:** Find the derivative of the function $f(a,d) = 4ab^3 + 9ad^2$ with respect to the variables involved. 2. **Identify variables and constants:** Here, $a$ and $d$ are variables, and $b$ is treated as a constant. 3. **Recall derivative rules:** - The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. - The derivative of a constant times a function is the constant times the derivative of the function. - When differentiating with respect to $a$, treat $d$ and $b$ as constants, and vice versa. 4. **Find partial derivatives:** - Derivative with respect to $a$: $$\frac{\partial}{\partial a}(4ab^3 + 9ad^2) = 4b^3 + 9d^2$$ - Derivative with respect to $d$: $$\frac{\partial}{\partial d}(4ab^3 + 9ad^2) = 0 + 9a \cdot 2d = 18ad$$ 5. **Final answer:** - $\frac{\partial f}{\partial a} = 4b^3 + 9d^2$ - $\frac{\partial f}{\partial d} = 18ad$