Would someone please take the partial derivatives for x and y for the following:

f(x,y) = [3x2y][(x+7)^(x/y)]

Thanks.

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- Sep 2nd 2012, 04:56 AMisaacchain rule with partial derivatives
Would someone please take the partial derivatives for x and y for the following:

f(x,y) = [3x2y][(x+7)^(x/y)]

Thanks. - Sep 2nd 2012, 05:02 AMProve ItRe: chain rule with partial derivatives
- Sep 2nd 2012, 05:23 AMisaacRe: chain rule with partial derivatives
i know about holding the other variables constant. i am getting tripped up on the x/y exponent as well as how to implement the chain rule. could you possibly work out the partial derivatives for x and y for me?

- Sep 2nd 2012, 05:48 AMProve ItRe: chain rule with partial derivatives
Yes I could, but I won't, as it goes against MHF policy. For another hint, when dealing with variables in your exponents, you should take the logarithm of both sides and simplify using logarithm rules, before differentiating implicitly.

- Sep 2nd 2012, 11:34 PMSworDRe: chain rule with partial derivatives
You can also use the fact that x^y = e^(y*ln(x))

- Sep 3rd 2012, 08:08 AMHallsofIvyRe: chain rule with partial derivatives
Or, use "logarithmic differentiation". With f(x,y) = [3x2y][(x+7)^(x/y)]= 6xy(x+7)^(x/y) (unless that "[3x2y]" means something completely different, like "[3x^3y]") we have

ln(f)= ln((6xy)(x+7)^(x/y))= ln(6)+ ln(x)+ ln(y)+ (x/y)ln(x+7)

Now, differentiating on both sides with respect to x, say,

(1/f) f_x= 1/x + (1/y)ln(x+7)+ x/(y(x+7) and we can solve for f_x by multiplying both sides by the original function, f.

The derivative with respect to y can be done similarly.