Prove the Power Rule (calculus)

[mattxr250]

Ok guys, I'm new here and I need some help with a math problem...

The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...

the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p and differentiate implicitly...assume that p and q are integers, where q>0.

Thanks for any help

Matt

 

[Fermat]

Are you allowed to assume that "d/dx[x^n] = nx^(n-1)" is true for n = an integer?

Have you differentiated "y^q = x^p" implicitly yet ?

 

[mattxr250]

yes I have and I came up with...


q(y^(q-1))(dy/dx) = p(x^(p-1))

but I'm stuck after I get here...i guess I could isolate dy/dx, but I'm not sure where to go from there...any help?

 

[Hurkyl]

Well, do you know what y is?

 

[Fermat]

if y = x^(p/q)
then
y' = (p/q)x^(p/q - 1)

Use the expression you got for for the implicit differentiatoin and the expression y^q = x^p and manipulatre them to end up with the required result, shown above.