Would someone please take the partial derivatives for x and y for the following:
f(x,y) = [3x2y][(x+7)^(x/y)]
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.
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.