Question

Use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Find \(\nabla f(x, y)\)

Short answer

The gradient of the function \(f(x, y) = 3 - \frac{x}{3} - \frac{y}{2}\) is \(\nabla f(x, y) = \langle -\frac{1}{3}, -\frac{1}{2} \rangle\).

Step-by-step solution

Find the partial derivative of \(f(x, y)\) with respect to \(x\)

The partial derivative of \(f(x, y)\) with respect to \(x\) is calculated by taking the derivative of \(f\) with respect to \(x\), while treating all other variables as constants. This gives \(f_x = -\frac{1}{3}\).

Find the partial derivative of \(f(x, y)\) with respect to \(y\)

The partial derivative of \(f(x, y)\) with respect to \(y\) is calculated in a similar way to the \(x\) partial derivative. This time, however, we take the derivative of \(f\) with respect to \(y\), while treating all other variables as constants. This gives \(f_y = -\frac{1}{2}\).

Write the gradient

The gradient of \(f(x, y)\) is written as a vector of the two partial derivatives calculated in Steps 1 and 2. So, \(\nabla f(x, y) = \langle f_x, f_y \rangle = \langle -\frac{1}{3}, -\frac{1}{2} \rangle\).