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Hi everyone I'm trying to prove the formula * by induction. I think I've done it right but I'm not sure about the last step. Is it right and if not how would I continue? Your help is appreciated.

*$\nabla u^n(x,y)=nu^{n-1}(x,y)\nabla u(x,y)$

Proof by induction: show true for n=1

$\nabla u(x,y)=u^{0}(x,y)\nabla u(x,y)$ $=\nabla u(x,y)$

Assume true for n=k

$\nabla u^k(x,y)=ku^{k-1}(x,y)\nabla u(x,y)$

Show true for n=k+1

$\nabla u^{k+1}(x,y)=ku^{k-1}(x,y)\nabla u(x,y) \nabla u(x,y)$

$\nabla u^{k+1}(x,y)=\nabla u^k(x,y)$$\nabla u(x,y)$=$\nabla u^{k+1}(x,y)$ (by inductive hypothesis)

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    Please include your work in your post written in Tex and not a picture. – 2017-02-11
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    Ok sorry about that – 2017-02-11
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    I'm a bit confused about your first line after 'Show true for $n=k+1$'. Shouldn't it be $$\nabla u^{k+1} = (k+1)u^{k} \nabla u$$ ? – 2017-02-11

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