Exercise 522Calculus › Multivariable calculus › Partial derivativesHigh schoolAdded 3 Nov 2026

Directional derivative

#directional derivative#gradient#direction

Statement

Let $f(x,y)=x^2y$ and $\vec u=(1,1)/\sqrt2$ . Compute the directional derivative $D_{\vec u}f(1,2)$ .

Hint

$D_{\vec u}f=\nabla f\cdot\vec u$ .

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Gradient. $\nabla f=(2xy,x^2)$ , so $\nabla f(1,2)=(4,1)$ .

Dot product. $D_{\vec u}f(1,2)=(4,1)\cdot\frac{(1,1)}{\sqrt2}=\frac{5}{\sqrt2}$ .

Conclusion. $D_{\vec u}f(1,2)=5/\sqrt2$ .