- #1

AwesomeTrains

Gold Member

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## Homework Statement

I want to find the partial derivatives in the point [itex](0,0)[/itex] of the function [itex]f:\mathbb R^2\rightarrow\mathbb R[/itex] [itex]

f(x,y):=

\begin{cases}

0 & \text{if } (x,y) = (0,0) \\

\frac{y^5}{2x^4+y^4} & otherwise

\end{cases}

[/itex]

## Homework Equations

Our definition of the partial derivatives in the direction [itex]\vec v = (v_1,v_2)[/itex] with [itex]\|\vec v \|_2 = 1[/itex] at the point [itex](0,0)[/itex]

[itex]D_v(f)(0,0)=\lim_{h\rightarrow 0} {\frac{f((0,0)+h\vec v)-f(0,0)}{h}}[/itex]

## The Attempt at a Solution

Straight forward:

[itex]D_v(f)(0,0)=\lim_{h\rightarrow 0} {\frac{f((0,0)+h\vec v)-f(0,0)}{h}}=\lim_{h\rightarrow 0} {\frac{f(h(v_1,v_2))}{h}}=\lim_{h\rightarrow 0} {\frac{\frac{h^5v_2^5}{2h^4v_1^4+h^4v_2^4}}{h}}= \frac{v_2^5}{2v_1^4+v_2^4}[/itex] Then in the direction [itex]\vec v = (0,1),(y-direction)[/itex] the tangent slope should be [itex]\frac{1^5}{2*0^4+1^4}=1[/itex]

Here's my problem: When I evaluate the same thing in maple I get 0. Where's my error?

(I've attached a picture of maple)

Also I can't see from the graph that the tangent slope in the y-direction should be 0, I think.

Any feedback is very appreciated :)

Alex