Thread: chain 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.

Originally Posted by isaac

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

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

Thanks.

Hint: When taking a partial derivative with respect to one variable, all other variables are treated as constants.

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?

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.

You can also use the fact that x^y = e^(y*ln(x))

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.