GATE 2025 ME – Question 46
The directional derivative of the function $f$ given below at the point $(1, 0)$ in the direction of $\frac{1}{2}(\hat{i} + \sqrt{3}\,\hat{j})$ is ____________ (rounded off to 1 decimal place).
$$f(x, y) = x^2 + xy^2$$
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Correct answer: 0.99 to 1.01
Explanation
The gradient is $\nabla f = (2x + y^2)\hat{i} + 2xy\,\hat{j}$, which at $(1, 0)$ is $2\hat{i} + 0\,\hat{j}$. The direction $\frac{1}{2}(\hat{i} + \sqrt{3}\hat{j})$ is already a unit vector. The directional derivative is $2 \times \frac{1}{2} = 1.0$.