Find the gradient vector field of the function $f(x,y)=x{e}^{9xy}$. Which of the following is the correct gradient vector field $\nabla f(x,y)$?

• A. $(e^{9xy}+9y{e}^{9xy},9x{e}^{9xy})$
• B. $(e^{9xy},9x{e}^{9xy})$
• C. $(e^{9xy}+9xy{e}^{9xy},9x^2{e}^{9xy})$
• D. $(e^{9xy},9y{e}^{9xy})$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the definition of the gradient vector field ∇f(x, y). The gradient is a vector composed of the partial derivatives of the function f(x, y) with respect to x and y. Specifically, ∇f(x, y) = (∂f/∂x, ∂f/∂y).
2. Step 2: Compute the partial derivative of f(x, y) = x e^{9xy} with respect to x. Treat y as a constant during this differentiation. Use the product rule: ∂f/∂x = ∂(x) * e^{9xy} + x * ∂(e^{9xy})/∂x.
3. Step 3: For the term ∂(e^{9xy})/∂x, apply the chain rule. Since e^{9xy} is an exponential function, its derivative with respect to x is e^{9xy} multiplied by the derivative of the exponent 9xy with respect to x. The derivative of 9xy with respect to x is 9y.
4. Step 4: Combine the results from Step 2 and Step 3 to find ∂f/∂x. This gives ∂f/∂x = e^{9xy} + 9xy e^{9xy}.
5. Step 5: Compute the partial derivative of f(x, y) with respect to y. Again, use the product rule: ∂f/∂y = ∂(x) * e^{9xy} + x * ∂(e^{9xy})/∂y. For ∂(e^{9xy})/∂y, apply the chain rule, where the derivative of 9xy with respect to y is 9x. Combine the results to find ∂f/∂y = 9x^2 e^{9xy}. The gradient vector field is therefore ∇f(x, y) = (e^{9xy} + 9xy e^{9xy}, 9x^2 e^{9xy}).