GATE 2024 CH – Question 53
Consider the function $f(x, y, z) = x^4 + 2y^3 + z^2$. The directional derivative of the function at the point $P(-1, 1, -1)$ along $(\hat{i} + \hat{j})$, where $\hat{i}$ and $\hat{j}$ are unit vectors in the $x$ and $y$ directions, respectively, rounded off to 2 decimal places, is _______
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Correct answer: 1.40 to 1.42
Explanation
The gradient is $\nabla f = (4x^3, 6y^2, 2z)$, which at $P$ is $(-4, 6, -2)$. The unit vector along $\hat{i} + \hat{j}$ is $\frac{1}{\sqrt{2}}(1, 1, 0)$. The directional derivative is $\frac{-4 + 6}{\sqrt{2}} = \sqrt{2} = 1.41$.