Directional Derivatives — Question 7

Question 7 – Solution

Strategy. Approach the origin along the ray $hu$ and keep the signed parameter $h$.

Step 1: Definition For $h\ne 0$, $$f(h\cos\theta,h\sin\theta) =\frac{h^3\cos^3\theta}{h^2} =h\cos^3\theta.$$ Therefore $$\boxed{D_{u}f(0,0)=\lim_{h\to 0}\frac{h\cos^3\theta}{h} =\cos^3\theta}.$$

Step 2: Gradient comparison Axis definitions give $f_x(0,0)=1$ and $f_y(0,0)=0$, so $$\nabla f(0,0)\cdot u=\cos\theta,$$ which differs from $\cos^3\theta$ for most angles.

Conclusion Every directional derivative exists, but its dependence on $u$ is not linear. Thus $f$ is not differentiable at the origin, and the gradient formula is not valid there.