Partial Derivative Homework: Show x\nablaf(x)=pf(x)

[ak123456]

Homework Statement

A function f: R^n--R is homogenous of degree p if f( $$\lambda$$x)=$$\lambda$$^p f(x) for all $$\lambda$$$$\in$$R and all x$$\in$$R^n
show that if f is differentiable at x ,then x$$\nabla$$f(x)=pf(x)

Homework Equations

The Attempt at a Solution

set g($$\lambda$$)=f($$\lambda$$x)
find out g'(1)
then how to continue ?

 

[ak123456]

any help?

 

[Office_Shredder]

$$g( \lambda _ = f( \lambda x)$$

Then $$g'( \lambda ) = \sum_{i=1}^{n} \frac{df}{dx_i} \frac{d( \lambda x}{dx_i}$$

The right hand side is obtained using the chain rule. Try to calculate what the right hand side really is

 

[ak123456]

$$g( \lambda _ = f( \lambda x)$$

Then $$g'( \lambda ) = \sum_{i=1}^{n} \frac{df}{dx_i} \frac{d( \lambda x}{dx_i}$$

The right hand side is obtained using the chain rule. Try to calculate what the right hand side really is

i don't know where is the formula for g' comes from

 

[HallsofIvy]

Office Shredder told you: it is the chain rule.